Download Fall Physics Activities - University of New Hampshire

In Search of Newton
Physics and Combined Materials
for Studio Calculus/Physics 1
University of New Hampshire
Dawn Meredith, Kelly Black, and Karen Marrongelle
with help from Mark Leuschner, Jim Ryan, Gertrud Kraut, Robert Simpson,
Joel Shaw, Mike Briggs,
Gagik Gavalian, Cyndi Heiner, and Yihua Zheng
and all the students who took this course!
Summer 2001
1 Support
provided by the National Science Foundation - NSF-DUE-9752485
2
Preliminary Document
University of New Hampshire
Contents
I
Kinematics
5
1 Describing Motion
7
2 Moving in two dimensions
13
3 Subtracting Vectors
19
4 Calculating Velocity and Acceleration
21
5 Projectile Motion Lab
25
6 Emily’s Walk
31
7 Emily’s walk - Redux
35
8 Kinematics Problem Solving
39
II
41
Dynamics
9 How do forces combine?
43
10 Investigating Force, Acceleration and Velocity
47
11 Banging Carts!
51
12 Third Law Pairs
53
13 Examining Forces
55
14 Thinking about friction
57
15 Atwood Machine Problem
59
16 Newton’s Method
63
17 Nasty Canasty vs. Monty Gue
67
3
4
CONTENTS
18 Moving in a Circle at Constant Speed
71
19 Forces and Circular Motion at Constant Speed
77
20 Drag Force on a Coffee Filter
79
III
87
Conservation Laws
21 Balance Point
89
22 Proof of Conservation of Momentum
97
23 Recognizing Conservation of Momentum
99
24 Air Drag and Euler’s method
101
25 Forces and work
105
26 Proof of Work-Energy Theorem
107
27 Proof of Conservation Laws
109
28 Contrast Work-Energy Theorem and Fnet = ma
111
29 Conservation Lab I
115
30 Conservation Lab II
121
IV
Rotational Motion
127
31 Rotational analogs to force and mass
129
32 Moment of inertia
133
33 Spinning our selves!
139
34 Moment of Inertia Experiment
143
35 Problem Using Torque and Moment of Inertia
145
36 Atwood Machine Problem - Redux
147
Preliminary Document
University of New Hampshire
Part I
Kinematics
5
Activity 1
Describing Motion
• Is it possible for t = 0? If so, what does it mean? If not, why not?
• Is it possible for t < 0? If so, what does it mean? If not, why not?
• Is it possible for t > 0? If so, what does it mean? If not, why not?
• Is it possible for x = 0? If so, what does it mean? If not, why not?
• Is it possible for x < 0? If so, what does it mean? If not, why not?
• Is it possible for x > 0? If so, what does it mean? If not, why not?
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ACTIVITY 1. DESCRIBING MOTION
Imagine that we have a set of ordered pairs which describe the motion of a ball:
(t1 , x1 ), (t2 , x2 ), (t3 , x3 ), . . .
Then we define ∆t ≡ ti − tj and ∆x ≡ xi − xj , where i and j are different integers.
∆ is therefore shorthand for change. Based on this definition, please answer the
following questions:
• Is it possible for ∆t = 0? If so, what does it mean? If not, why not?
• Is it possible for ∆t < 0? If so, what does it mean? If not, why not?
• Is it possible for ∆t > 0? If so, what does it mean? If not, why not?
• Is it possible for ∆x = 0? If so, what does it mean? If not, why not?
• Is it possible for ∆x < 0? If so, what does it mean? If not, why not?
• Is it possible for ∆x > 0? If so, what does it mean? If not, why not?
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University of New Hampshire
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We define average velocity v̄ ≡ ∆x/∆t. Based on this definition, please answer the
following questions:
• In words, what does it mean if v̄ = 0?
• Give two different examples (in words or with a plot) for which v̄ = 0.
• In words, what does it mean if v̄ < 0?
• Give two different examples (in words or with a plot) for which v̄ < 0.
• In words, what does it mean if v̄ > 0?
• Give two different examples (in words or with a plot) for which v̄ > 0.
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Calculus and Physics
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ACTIVITY 1. DESCRIBING MOTION
We can represent x(t) (the location of an object in time) in several ways, as a table,
as a plot, in words, as a stroboscopic picture (dots at the locations at t=1sec, 2
sec, 3 sec,...) or as an equation. Given the data table below,
t (seconds)
0
1
2
3
4
5
6
x (meters)
1
2
9
28
28
9
2
28
24
20
16
12
8
4
4
2
6
8sec
• Sketch the motion on the grid provided.
• Given that v̄ = ∆x/∆t, how can you estimate v̄ on the plot?
• Between what two points is the v̄ the greatest?
• Describe the motion in words. That is, during what intervals is it moving leftward,
rightward, slowing down, or speeding up?
• Give a stroboscopic version of the motion for t = 0 to t = 3 seconds on the plot
below. That is, put a dot at the location of the particle at t = 0, 1, 2, 3 seconds.
0
4
Preliminary Document
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12
16
20
24
28 m
University of New Hampshire
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• Below are some plots of x(t). Some of the following plots depict motion that cannot
occur in the real world. Beneath each plot, either describe the motion in words (if
it can occur) or tell why it cannot occur.
x
x
t
x
x
t
In Search of Newton
t
t
Calculus and Physics
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Preliminary Document
ACTIVITY 1. DESCRIBING MOTION
University of New Hampshire
Activity 2
Moving in two dimensions
Today we will work on motion in two-dimensions, along a flat surface. You will see
quickly that trigonometry is important, so we begin with a quick review of trigonometry.
θ
c
a
ϕ
b
• Given the right triangle above, if a = 3m and b = 2m, what is c?
• If a = 3m and b = 2m, what are the angles θ and φ?
• What is cos θ in terms of the sides a, b, c?
• What is sin θ is terms of the sides a, b, c?
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ACTIVITY 2. MOVING IN TWO DIMENSIONS
Imagine a child Emily taking a short walk starting from her front porch. For each of
the following situations, calculate two things:
• the total distance that Emily walked,
• how far Emily is from her front porch at the end of the walk, and in what direction.
For each case, draw a sketch of the path she took (take 1 grid= 1 foot). Don’t simply
estimate the distances from the plot, instead use trigonometry to find the exact values
of the distances asked for above.
1. She walks 3 feet east, then 4 more feet east.
North
West
East
South
2. She walks 3 feet east, then 4 feet west.
North
West
East
South
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University of New Hampshire
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3. She walks 3 feet east, then 4 feet north.
North
West
East
South
4. She walks 3 feet west, then 4 feet south.
North
West
East
South
5. She walks 3 feet east, then 4 feet northeast.
North
West
East
South
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ACTIVITY 2. MOVING IN TWO DIMENSIONS
6. She walks 3 feet 30 degrees north of east, then 4 feet 80 degrees north of east.
North
West
East
South
7. Look back on the work you have done to this point.
(a) What procedure have you used to calculate the total distance traveled?
(b) What procedure have you used to calculate the distance she is from her front
porch?
(c) If you haven’t done so already, state a procedure that will work that does not
rely on drawing a sketch. Then use this method to calculate the distance from
her front porch in the next situation.
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University of New Hampshire
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8. She walks 3 feet 30 degrees south of west, then 4 feet northeast.
North
West
East
South
9. In your above procedure, how do you differentiate distances traveled east from those
traveled west? North from south?
10. Can you easily calculate the distance from her front porch using the sketch?
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Calculus and Physics
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Preliminary Document
ACTIVITY 2. MOVING IN TWO DIMENSIONS
University of New Hampshire
Activity 3
Subtracting Vectors
1. What number do you need to add to 3 to get zero? (This is known as the inverse1
of 3.)
2. Verify that subtracting 3 from any number (say 5) is the same thing as adding the
additive inverse of 3.
3. What vector do you need to add to 3 feet east to get zero net displacement? (This
is the inverse of the vector 3 feet east.)
4. In general, how would you get the additive inverse of a vector? (Explain both in
pictures and in words.)
5. How are the components of the original vector and its additive inverse related?
That is, how do their magnitudes compare? Their signs?
1
or more correctly the additive inverse, since 3 also has a multiplicative inverse.
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ACTIVITY 3. SUBTRACTING VECTORS
~ from A
~ is the same as
6. By analogy with subtraction of numbers, subtraction of B
~
~
~
~ where A
~ is
adding the additive inverse of B to A. Using this rule, calculate A − B
~ is 4 feet 80 degrees south of west.
3 feet 30 degrees north of east and B
Do the subtraction both graphically and numerically, and feel free to use numbers you calculted on the previous worksheet. Compare your answers using both
methods. Are they reasonably close?
North
West
East
South
Preliminary Document
University of New Hampshire
Activity 4
Calculating Velocity and
Acceleration
In this activity you will analyze ticker tapes from a cart moving up or down a ramp.
These tapes were made by attaching the tape to the moving cart. Every 1/60 of a second
there was a strike that made a dot on the tape.
Using a ruler, a calculator, and the definition of average acceleration and velocity, find
the average acceleration of the moving object for each time interval. Use the table on
the back of this sheet to organize your work. Express your answer in meters per second
squared.
Note that in a previous experiment, we have let the computer calculate the velocity
and acceleration given the position at equal times. This activity allows you to see the
details of that calculation.
Use the space below to outline your procedure. Check the procedure with an instructor before continuing.
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ACTIVITY 4. CALCULATING VELOCITY AND ACCELERATION
Column 1 Column2 Column 3 Column 4 Column 5 Column 6
Data
Description
Units
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University of New Hampshire
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On the axes below, plot x, v, and a.
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Calculus and Physics
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ACTIVITY 4. CALCULATING VELOCITY AND ACCELERATION
Preliminary Document
University of New Hampshire
Activity 5
Projectile Motion Lab
1. Purpose Projectile Motion - the motion of an object moving in the air near the
surface of the earth - is one motion that has numerous applications in everyday life.
In this lab we will measure the position of a ball thrown in the air, let the computer
calculate (using an approximate scheme) the velocity and acceleration. Then we
will model the position, velocity and acceleration using mathematical formulas, and
interpret the parameters in the models.
2. Setup the experiment:
(a) Turn on the computer and launch Videopoint.
(b) Open the movie Pasco 104.
(c) On the next screen you need to tell the program how many objects to follow.
For movie 104 you have only one point; enter the number 1 and click okay.
(d) Enlarge the movie by clicking on the leftmost button in the upper right corner
of the movie window.
3. Expectations: Let’s assume that the x and y motions are independent of each other.
(a) What acceleration do you expect in the x-direction?
ax =
(b) What acceleration do you expect in the y-direction? What value is it? Does
it change with time or is it constant?
ay =
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ACTIVITY 5. PROJECTILE MOTION LAB
(c) Given the accelerations in the last two questions, use anti-derivatives to obtain
the velocity in each direction.
vx =
vy =
(d) Given the velocities in the last two questions, use anti-derivatives to find the
position in each direction.
x=
y=
4. Run the “experiment” and take data:
(a) Run the movie by clicking on the button on the lower left next to the scroll
bar. Note that you can step through the movie using the buttons to the right
of the scroll bar.
(b) Place the movie at the beginning by dragging the scrollbar all the way to left.
(c) On each frame, place and click the cursor directly over the middle of the
launched ball. This places the location information in a table. Be sure to click
on the same place on the object in each frame. After you have done this for
each frame in the movie, you can go back to any frame and adjust the location
of the cursor by dragging it with your mouse, or by using the “nudge” tool
(the set of four arrows) to move small amounts in any direction.
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University of New Hampshire
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(d) Before the data can be analyzed, we need to set the scale for the data by
letting it know how many pixels are a meter for this movie. You can do the
scaling in the following steps.
i. Under the ”Movie” menu, select ”scale movie”.
ii. The first dialog box should have the known length to be 1 meter, and the
scale “fixed”. Click on continue.
iii. You will then be instructed to click on one end of a known length: use
your cursor to click on one edge of the meter stick in the lower portion of
the movie frame.
iv. Now you need to click on the other end of the length: click on the other
end of the meter stick. Now the computer knows the conversion from
pixels to meters and the data is automatically converted.
5. Graph and model the data:
(a) Click on the graph icon on the left of the screen (or choose ”new graph” under
the ”view” menu).
(b) The first coordinate is time. Keep it as time.
(c) The second coordinate is x. Select position, velocity and acceleration underneath and then click ”okay”. The plot should pop up. Verify that the units
are in meters and seconds.
(d) Next you should fit the data. First, click on the x(t) plot to select it, then
under the Graph menu, chose Add/Edit Fit. Fit x(t) to appropriate function
that you choose above in the expectations section.
(e) Repeat for vx (t) and ax (t) plots. If the equations don’t model the data well,
stop and think about why there is disagreement.
(f) Repeat the above steps for the motion in the y-direction.
(g) Print out your plots.
6. Do the fits for x, vx and ax seem to accurately model the data? Explain.
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Calculus and Physics
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ACTIVITY 5. PROJECTILE MOTION LAB
7. In the space below write out both the fit from the computer and your original
expectation for ax , vx and x.
prediction
fit
x
vx
ax
8. By comparing the prediction with the fit, find the value of the acceleration from
x, vx and ax fits. Write those below. Are they in good agreement?
9. By comparing the prediction with the fit, find the value of the initial velocity from
x, vx fits. Write those below. Are they in good agreement?
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University of New Hampshire
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10. In the space below write out both the fit from the computer and your original
expectation for ay , vy and y.
prediction
fit
y
vy
ay
11. By comparing the prediction with the fit, find the value of the acceleration from
y, vy and ay fits. Write those below. Are they in good agreement?
12. By comparing the prediction with the fit, find the value of the initial velocity from
y, vy fits. Write those below. Are they in good agreement?
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Calculus and Physics
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Preliminary Document
ACTIVITY 5. PROJECTILE MOTION LAB
University of New Hampshire
Activity 6
Emily’s Walk
During our last class we worked with displacement (∆~x ≡ ~x2 − ~x1 ) and found that it was
a vector. On this worksheet we will work with displacement, velocity and acceleration
and learn about their vector properties.
Constant speed Emily walks 6 meters east and then 6 meters north-east. She walks
at a constant speed of 2 meters per second for the whole trip and begins walking at t = 0
seconds.
1. In lecture we will discuss what is her final position (that is, what is her displacement) after the whole trip? Draw a picture using vectors and state the result in
both component form (with î and ĵ notation) and as magnitude and direction.
North
East
2. First leg of the trip
(a) What is her eastward velocity on the first leg of the trip?
(b) What is her northward velocity on the first leg of the trip?
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ACTIVITY 6. EMILY’S WALK
(c) Express her velocity on this leg of the trip in î and ĵ notation.
(d) Use your three previous answers and anti-derivatives to write down her position in vector notation for her first leg of the trip.
(e) What is the domain of this function? What is the range? How does it relate
to your sketch?
3. Second leg of the trip
(a) What is her eastward velocity on the second leg of the trip?
(b) What is her northward velocity on the second leg of the trip?
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University of New Hampshire
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(c) Express her velocity on this leg of the trip in î and ĵ notation.
(d) Use antiderivatives to find her position in vector notation for the second leg
of the trip. Be very careful as you evaluate the constants.
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ACTIVITY 6. EMILY’S WALK
4. Whole trip
(a) Calculate her northward and eastward average velocities for the whole trip.
(b) Express her average velocity for the whole trip in î and ĵ notation.
(c) Find the magnitude and direction of her average velocity for the whole trip.
(d) How does the direction of the average velocity compare with the direction of
her total displacement? are they the same or different?
(e) Write down the equation that relates average velocity to total displacement.
Discuss with your group the answer to your last question in light of the definition.
Preliminary Document
University of New Hampshire
Activity 7
Emily’s walk - Redux
Non-constant speed This is an exercise in working with displacement, velocity and
acceleration when speeds are not constant. Emily decides to get a bit more tricky on
you. She goes for a walk, and her coordinates are changing in time:
x(t) = 2t m/s − t2 m/s2 ,
y(t) = t2 m/s2 .
1. Verify that the following graph is the path that she takes by calculating her location
at t = 0, 1, 2 and 3 second, plotting these locations as large circles and verifying
that they lie on the curve.
t
x
y
0 sec
1 sec
2 sec
3 sec
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ACTIVITY 7. EMILY’S WALK - REDUX
Path Taken By Emily
9
8
7
Y Position
6
5
4
3
2
1
0
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
X Position
0
0.5
1
1.5
In all of the following, give your answers in vector notation, that is, using î and ĵ.
2. What is the average rate of change of her position from t = 1 to t = 2?
3. What is the average rate of change of her position from t = 1 to t = 1.5?
4. What is the average rate of change of her position from t = 1 to t = 1.25?
5. You have just calculated the first three terms in a sequence. Does the sequence
appear to be converging? If so, to what vector?
6. What is her velocity vector at any given time?
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University of New Hampshire
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7. Calculate her velocity vector at t = 0, 1, 2 and 3 seconds.
8. Sketch the velocity vectors on the plot. In order to do this, you must pick a scale:
1 meter on the plot is equal to how many meters/second?
9. What is her acceleration vector as a function of time?
10. Calculate her acceleration vectors at t = 0, 1, 2 and 3 seconds.
11. Sketch her acceleration vectors on the plot. What scale will you use?
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Calculus and Physics
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Preliminary Document
ACTIVITY 7. EMILY’S WALK - REDUX
University of New Hampshire
Activity 8
Kinematics Problem Solving
1. You are helping to design a machine that will acclerate electrons using charged
plates at the end of long hollow tubes. As the electrons move between the plates,
the charge on the plates increases, so the acceleration of the electrons increases
with time as follows: a(t) = 5 × 106 t3 . If the electron starts with an initial velocity
of zero, how long does it take for the electron to travel .5 meters between the
plates? (You look up information about electrons and find that they have a mass
of 9.11 × 10−31 kg and a charge of 1.6 × 10−19 Coulombs.)
-
a
+
+
+
v=0 at left edge
2. (Note: this situation is a bit contrived, but the solution is still instructive!) There is
a bottomless pit in the basement of the physics building (in the Special Equipment
room). Inside this specially constructed pit (lined with kryptonite). This kryptonite
changes the accleration due to gravity from the usual 9.8 m/s2 downward to a highly
suprising 2t m/s2 upward, where t = 0 is the instant something is thrown into
the pit. You toss an apple downward into the pit at t = 0s with a velocity of
4m/s. Take the initial position to be zero. Considering this to be a one-dimensional
problem, find expression for the velocity and position of the apple at any given time
(as long as it remains in the pit). Does the apple ever come back to you or does it
keep going?
3. You have a toy rocket that you shoot up in the air. For the first four seconds while
the rockets are firing, they provide an acceleration of 21 m/s2 . After four seconds
the rocket cuts out and only the acceleration due to gravity is present. How high
will the rocket go and how long will it take to reach to top of its motion?t
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ACTIVITY 8. KINEMATICS PROBLEM SOLVING
University of New Hampshire
Part II
Dynamics
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Activity 9
How do forces combine?
The question we want to answer in this exercise is how to add forces in order to get a
net force of zero and no acceleration.
Equipment: The instructor will demonstrate the force table which we will use in the
following activity.
1. For each of the following three scenarios, make an experienced guess at which angle
you would hang an object to balance the ring. What should the mass of that object
be? Sketch in your answer and verify by pulling the ring that it is balanced.
50gm
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ACTIVITY 9. HOW DO FORCES COMBINE?
50gm
100 gm
50gm
100gm
100 gm
100gm
2. Did increasing the mass of the hanging objects in the x direction change the mass
of the hanging object needed in the y direction?
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University of New Hampshire
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3. For the following scenarios, make an reasoned guess (based on your answers from
the last page) at which angle you would hang an object to balance the ring. What
should the mass of that object be? Hint: if your could hang two weights in order
to balance, where would you hang them and how big would they be? How can you
combine those two weights into an equivalent single weight?
100gm
100gm
In Search of Newton
100 gm
50 gm
Calculus and Physics
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ACTIVITY 9. HOW DO FORCES COMBINE?
4. Explicitly as you can, describe how you have calculated the necessary mass and
angle needed to balance the given objects. Be as general as possible.
5. Use the method that you just described to find the angle and mass of a balancing
object in the following case:
50 gm
75 deg
100gm
Preliminary Document
University of New Hampshire
Activity 10
Investigating Force, Acceleration
and Velocity
• Consider a cart being pushed along a horizontal table at constant velocity. Sketch
below at three different times all of the forces you belive are acting on that cart. Be
sure to indicate the relataive size of those forces by the relative size of the vectors1
.
• Discuss as a group what forces are acting. Modify your diagrams above if your
group members mention other forces that you believe are acting on the cart.
• Discuss as a group how these forces combine to make the cart move at constant
velocity. Write your group conclusions below.
1
This activity is based on “Teaching Introducotry Physics to College Students” by Dewey Dykstra
in Constructivism: Theory Perspectives, and Practice, Teachers College, 1996 and “Explaining the ‘at
rest’ condition of an object” by Jim Minstrell in The Physics Teacher 20 pages 10-16.
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48 ACTIVITY 10. INVESTIGATING FORCE, ACCELERATION AND VELOCITY
• Consider a cart being pushed along a horizontal table at constant acceleartaion.
Sketch below at three different times all of the forces you belive are acting on that
cart. Be sure to indicate the relataive size of those forces by the relative size of the
vectors.
• Discuss as a group what forces are acting. Modify your diagrams above if your
group members mention other forces that you believe are acting on the cart.
• Discuss as a group how these forces combine to make the cart move at constant
accleration. Write your group conclusions below.
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• Consider a cart on a horizontal table being pulled by an object which hangs over
the table. Sketch below the force, accleration, velocity, and position of the cart on
the table. (Ignore the scales on the plots!)
A
1
1
0.8
0.8
acceleration
force
B
0.6
0.4
0.2
0
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
1
1
0.8
0.8
0.6
0.6
0.4
0.2
0
0.6
0.8
1
0.6
0.8
1
time
position
velocity
time
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0
0
time
0.2
0.4
time
• Explain why you drew your predictions as you did.
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50 ACTIVITY 10. INVESTIGATING FORCE, ACCELERATION AND VELOCITY
• Procedure: Now you can check the predictions that you made on the last page.
• Preparing the equipment:
– Getting the track ready:
height.
Make sure the track is level and at a comfortable
– Getting the equipment: Make sure you have a cart with an attached force
sensor, a long string, a weight hanger and three or four small weights (about
50 gm each).
– Preparing the computer and force probe: Turn on the computer and the ULI
(universal lab interface). Plug the force probe into Din 1. Launch MacMotion.
Display four graphs at once (position, velocity, acceleration, and force). Make
the graphs large!
– Calibrating the Probe: First, make sure the probe is set at the 10 N reading
(look for the switch at the top). Then, under the Collect menu, choose
Calibrate Force Probe (at the bottom), and then Calibrate Now. It will
ask you to remove all force from the probe. Do so. Then it will ask you to
apply a known force. Hang a 200 gm weight (don’t forget to include the mass
of the hanger), with a force of 1.962 Newtons. Type this value in. [You can
check the calibration by hanging a different weight, and clicking on start. The
force plot should show the appropriate constant value.]
– Make sure that the motion detector is plugged into the ULI and is properly
positioned to take data.
• Data runs: Take data for at least two different values of the hanging mass. Make
sure that the data makes sense (e.g. is the x(t) plot reasonable)? Print out the
plots.
• What can you conclude from these experiments about how forces combine to give
constant acceleration? to give constant velocity? If your experiment does not
agree with your predictions, how must you modify your understanding of forces
and acceleration and velocity to match the data?
Preliminary Document
University of New Hampshire
Activity 11
Banging Carts!
Consider two carts which are moving in opposite directions (or perhaps one is sitting
still) and they bang into each other.
1. Think of two situations in which you expect the force of cart A on cart B to be the
same as the force of cart B on cart A (you can vary things such as initial speeds
and masses of the cars). Carry out the experiment and see if your predictions are
indeed correct. Write down your predictions (before the experiment!) and your
results.
2. Think of two situations in which you expect the force of cart A on cart B to
be different from force of cart B on cart A. Carry out the experiment and see
if your predictions are indeed correct. Write down your predictions (before the
experiment!) and your results.
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Preliminary Document
ACTIVITY 11. BANGING CARTS!
University of New Hampshire
Activity 12
Third Law Pairs
elevator
Consider a book lying on a table. In the cases where the book
is moving, imagine that the book and table are in a moving
elevator, and the book and table are not moving with respect
to one another.
For each of the situations below,
• Draw the free body diagram for the book which includes
all of the forces acting on the book, shows their direction
and relative magnitude.
book
• Calculate the magnitude of each force knowing that the
mass of the book is 4 kg (take g = 10 m/s2 ).
table
1. The velocity of the book and table is zero, and the acceleration is zero.
2. What is the third law pair for each force acting on the book? Explain.
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ACTIVITY 12. THIRD LAW PAIRS
3. The v(0) = 3 m/s upward and the acceleration is zero (the table and book are in
a moving elevator!)
4. The v(0) = 3 m/s upward and a = 2 m/s2 upward.
5. The v(0) = 3 m/s upward and a = 10 m/s2 downward.
6. In each case above, was the normal force equal to the weight of the book? If not,
how did you find the magnitude of the normal force?
Preliminary Document
University of New Hampshire
Activity 13
Examining Forces
You push a 5 kg block along a frictionless table with a force of 22 N.
1. Draw the free body diagram of the block. Be sure to denote what object exerts the
force (first subscript), which object feels the force (second subscript) and the type
of force (N=normal, f=frictional, G=gravitational, T=tension, etc.)
2. Find the acceleration of the block.
22 N
5 kg
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ACTIVITY 13. EXAMINING FORCES
Now you realize that the 5 kg block is actually made up of a .5 kg, 1.5 kg and 3 kg
block.
1. Draw the free body diagram of each block. Be sure to denote what object exerts
the force (first subscript), which object feels the force (second subscript) and the
type of force (N=normal, f=frictional, G=gravitational, T=tension, etc.)
2. Find the acceleration of each block.
3. Is your answer for the accelerations sensible? That is, do you expect the accelerations to be the same or different for all three blocks?
.5
Preliminary Document
1.5
3kg
5 kg
22 N
University of New Hampshire
Activity 14
Thinking about friction
For each object in each sitatuion, determine the type of friction force acting on it due to
each surface it touches1 . Write S for static, K for kinetic, or zero if there is no friction
acting at all. For kinetic friction only, give the direction. (Usually the direction of static
friction cannot be determined without using Newton’s Laws.)
A
Two blocks are arranged as
shown. The upper block remains at rest. The lower
block moves to the right.
B
1. Both blocks are at rest. The string connecting them is slack.
(a) Block A: static, kinetic (give direction), or none?
(b) Block B upper surface: static, kinetic (give direction), or none?
(c) Block Blower surface: static, kinetic (give direction), or none?
1
From the work of the Physics Education Research Group at the University of Massachusetts at
Amherst
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ACTIVITY 14. THINKING ABOUT FRICTION
Two blocks are attached to
the same piece of string as
shown. A rope under tension is attached to the lower
block.
A
B
1. Both blocks are at rest. The string connecting them is slack.
(a) Block A: static, kinetic (give direction), or none?
(b) Block B upper surface: static, kinetic (give direction), or none?
(c) Block Blower surface: static, kinetic (give direction), or none?
2. Both blocks are at rest. The string connecting them is taut.
(a) Block A: static, kinetic (give direction), or none?
(b) Block B upper surface: static, kinetic (give direction), or none?
(c) Block Blower surface: static, kinetic (give direction), or none?
3. Both blocks are moving at the same speed. The string connecting them is taut.
The lower block moves to the right.
(a) Block A: static, kinetic (give direction), or none?
(b) Block B upper surface: static, kinetic (give direction), or none?
(c) Block Blower surface: static, kinetic (give direction), or none?
Preliminary Document
University of New Hampshire
Activity 15
Atwood Machine Problem
In the figure below there is a cart with mass M which moves without friction over the
table top. It is attached to a cord that wraps over a frictionless pulley to a second block
with mass m. The cord and pulley are massless. The block falls and accelerates the cart
to the right. What is the acceleration of the hanging block? What is the tension in the
cord?
(Note: if you find working with variables M and m a bit confusing, try using values
first. But then redo the problem with variables.)
A
B
1. Gather:
(a) What values are you given?
(b) What values do you need to find?
(c) Draw a free body diagram for both the cart and the block. Label all forces
with type of force, body exerting the force, body feeling the force.
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ACTIVITY 15. ATWOOD MACHINE PROBLEM
(d) What expectations about the acceleration of the block? That is it equal to
the acceleration due to gravity? Less? More?
(e) Would the acceleration of the block increase or decrease if the mass of the cart
increases?
(f) Would the acceleration of the block increase or decrease if the mass of the
block increases?
(g) Would the tension in the cord increase or decrease if the mass of the cart
increases?
2. Organize: What general approach(es) would you use here?
(a) Can you use average acceleration = change in velocity over change in time?
Explain.
(b) Can you use that acceleration is the derivative of v(t)? Explain.
(c) Can you use that Fnet = ma? Explain.
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3. Analyze: Now that you have a procedure, use that procedure to find your unknowns.
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ACTIVITY 15. ATWOOD MACHINE PROBLEM
4. Learn and Check:
(a) First check: Are the units on your answer correct?
(b) Does your answer agree with your expectations?
(c) Do limiting cases make sense?
i. That is, what is the acceleration if M =0, and does this make sense?
ii. If m=0?
iii. If g=0?
(d) Can you do this problem another way? Can you see the answer all at once?
(e) Why did we ask you to do this problem?
Preliminary Document
University of New Hampshire
Activity 16
Newton’s Method
35
30
25
f
20
15
10
5
0
−5
0
0.5
1
1.5
2
2.5
3
3.5
4
t
1. Sketch the tangent line to the curve at t = 3.5.
2. Approximate the time when the tangent line crosses the t-axis.
3. Sketch the tangent line to the curve for this new time value.
4. Approximate the time when this new tangent line crosses the t-axis.
5. Sketch the tangent line to the curve for this new time value.
6. Approximate the time when this new tangent line crosses the t-axis.
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ACTIVITY 16. NEWTON’S METHOD
7. Sketch the tangent line to the curve for this new time value.
8. Approximate the time when this new tangent line crosses the t-axis.
9. What value of time do you approach? What is special about this time value?
Preliminary Document
University of New Hampshire
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Find the general formula:
1. Given t0 , f (t0 ), and f 0 (t0 ), find the formula for the tangent line using the slopeintercept form for a line.
2. Find the value of t at which the tangent line crosses the t-axis. This equation is
the basis of Newton’s method.
3. Now we want you to find the root of f (t) = t3 − 1.4t2 − .9t − 3.6 :. We give you an
initial guess of t0 = 3.5. You will find that Matlab is a great help here. Below is
some sample code to get you started. Use the up arrow to recall previous equations
so you need not retype the formulas for each iteration.
t0=3.5
f=t0^3-1.4*t0^2-.9*t0-3.6
fp=[put your derivative here]
t=t0-f/fp
t0=t
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Fill in the
table going
across the
rows. Once
tn+1 is
found, use
this new
value for the
second
column of
the following
row.
ACTIVITY 16. NEWTON’S METHOD
4. Fill in the following table for f (t) = t3 − 1.4t2 − .9t − 3.6 :
n
0
tn
3.5
f (tn )
f 0 (tn )
tn+1 = tn −
f (tn )
f 0 (tn )
1
2
3
4
5
Preliminary Document
University of New Hampshire
Activity 17
Nasty Canasty vs. Monty Gue
Archvillian Nasty Canasty is in his hand-car moving at 50 metres per second eastward
when he notices that his nemesis, Monty Gue, is traveling straight at him, moving westward, on the same set of tracks. Monty is in a bullet-proof train and is moving at a
constant speed of 50 metres per second. Nasty Canasty orders his slow moving henchmen to reverse the direction. As they do so, they apply an acceleration of t/2 m/s3
westward. The acceleration is applied when Monty is only 1000 meters away from Nasty.
Will Nasty escape?
1. Gather information - remember, do not do any calculations yet!
• What information are you given?
• Draw a picture of the situation and label your coordinate system.
• Qualitatively describe and sketch the motion of both vehicles. Based on these
sketches, is it possible that Nasty escapes? Is it possible that he doesn’t?
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ACTIVITY 17. NASTY CANASTY VS. MONTY GUE
• Does there appear to be any missing information? Any extraneous information?
• What is the question asking? That is, what equation is the question asking
us to solve?
2. Organize - don’t do any calculations yet!
• What general approach will you use to solve this (e.g. Fnet=ma? Estimating
x from the plot of a? something else?) To decide which approach is best, find
the method connects what you know with what you want to find out.
Preliminary Document
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3. Use the approach on the last page to find the answer to the question.
• Setup the equation(s) you need to solve.
• You should now see that you need to use Newton’s method to solve this problem. What equation do you want to find the root of?
• What is the derivative of that equation?
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ACTIVITY 17. NASTY CANASTY VS. MONTY GUE
• In order to find the root, what is a good guess for t0 ? Hint: When would they
hit if Nasty didn’t accelerate? Will hitting occur before or after this?
• Use Matlab to find the root of this equation.
– After the first iteration, be sure to use the up arrow key so you don’t have
to retype the same commands each iteration.
– Monitor the value of the function we are finding the roots of. Is it taking
on reasonable values?
– When is your answer close enough? Hint: think about significant digits!
– Write down the numbers below for each iteration:
n
0
tn
f (tn )
f 0 (tn )
tn+1 = tn −
f (tn )
f 0 (tn )
1
2
3
4
5
4. Check.
• Are your units on the answer correct?
• Does your answer seem reasonable?
• Can you solve it another way to check?
• Why did we ask this question?
Preliminary Document
University of New Hampshire
Activity 18
Moving in a Circle at Constant
Speed
In this worksheet we will consider the details of moving in a circle at constant speed.
• Recalling the tutorial on motion in two dimensions, sketch the velocity and acceleration vectors at the three points marked below:
A
B
C
• The term “centripetal” means center-seeking, “centrifugal’ means center-fleeing.
Does either work describe the acceleration you sketched above? If so, which one?
• We have the direction of the acceleration determined, now we turn to learning about
the magnitude. Before we do this, we need to recall how to calculate acceleration
graphically. Below are two velocity vectors at two different times. Use those vectors
to graphically calculate the acceleration (at least the direction). (Hint: what is the
definition of average acceleration?)
v1
v2
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ACTIVITY 18. MOVING IN A CIRCLE AT CONSTANT SPEED
• On the next page are two circles of different radii on which two particles travel with
the same speed. One radius is twice the size of the other. In the section we will see
how the radius affects the magnitude of the acceleration.
– Pick two points on each circle that are separated by the same ∆t. (Separate
them by about an 1/8 of a circle on the smaller one.) How can you be sure
that the ∆t is the same for both circles?
– At each of the four points, sketch the velocity vector. Lengths of velocity
vectors must have the correct relative length. What lengths (in inches or
centimeters) are you choosing for each velocity vector? (The drawings will be
clearer if your velocities are about 1 or 2 inches long.)
∗
∗
∗
∗
v1small =
v2small =
v1big =
v2big =
– Use those velocity vectors to calculate the average acceleration for each circle.
How big is each acceleration (in inches or centimeters)?
∗ asmall =
∗ abig =
– How can we be sure that we are fairly comparing magnitudes of the two
acceleration vectors? That is, how do we know the scale that we used for both
is the same?
– State in words why the magnitude of the radius must affect the magnitude of
the acceleration.
– Is what we’ve seen here consistent with what you learned about the effect of
curvature on acceleration in the Motion in Two Dimensions tutorial?
Preliminary Document
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In Search of Newton
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ACTIVITY 18. MOVING IN A CIRCLE AT CONSTANT SPEED
• On the next page are two circles of the same radius on which two particles travel
with different speeds. One speed is twice the magnitude of the other. In this section
we will see how the speed affects the magnitude of the acceleration.
– Pick two points on each circle that are separated by the same ∆t. (Separate
the points on the faster one by about an 1/8 of a circle.) How can you be sure
that the ∆t is the same for both circles?
– At each of the four points, sketch the velocity vector. Lengths of velocity
vectors must have the correct relative length. What lengths (in inches or
centimeters) are you choosing for each velocity vector? (The drawings will be
clearer if your velocities are about 1 or 2 inches long.)
∗
∗
∗
∗
v1fast =
v2fast =
v1slow =
v2slow =
– Use those velocity vectors to calculate the average acceleration for each circle.
How big is each acceleration (in inches or centimeters)?
∗ afast =
∗ aslow =
– State in words why the magnitude of the velocity must affect the magnitude
of the acceleration.
Preliminary Document
University of New Hampshire
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ACTIVITY 18. MOVING IN A CIRCLE AT CONSTANT SPEED
• The book gives a proof (see section 4.7) that the acceleration of a particle moving
in a circle at constant speed is equal to v 2 /r.
– Are the units of this correct?
– Is this formula qualitatlively consistent with your results? That is, does the
acceleration increase if the velocity increases? Does the acceleration decrease
if the radius increases?
Preliminary Document
University of New Hampshire
Activity 19
Forces and Circular Motion at
Constant Speed
This activity focuses on forces and motion in a circle; that is, what can cause motion in
a circle at constant speed? Consider the following situations: For each scenario below,
draw a free body diagram and be sure to include the direction of the acceleration, if
there is any. Include only forces that we have worked with already (e.g. normal, tension,
weight, friction).
• A ball swung on a string in a vertical circle, at the top and bottom of the swing
• A ball swung on a string in a cone (the string is between horizontal and vertical).
• A person in an amusement park ride, the one that spins fast and then drops the
floor out the bottom
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ACTIVITY 19. FORCES AND CIRCULAR MOTION AT CONSTANT SPEED
• A car moving in a circle on a flat track.
• A car moving in a circle on a banked track with no friction. (Yes, the car would
slip off the track if it were not moving. Consider only the case when it is already
moving!)
Looking back over these situations,
• Are the forces completely balanced in each case? How do you know?
• We often here the term centripetal force for something that moves in a circle. What
does that term mean to you, either from past experience or from what you have
just done in this activity?
Preliminary Document
University of New Hampshire
Activity 20
Drag Force on a Coffee Filter
1. If you consider gas molecules in the air to be small hard spheres, how would you
explain or describe the force known as drag force or air resistance?
2. Consider a coffee filter falling. What features of the physical situation should affect
the magnitude of the force? Are they directly or indirectly proportional to the
force? Explain! (Many of you may be familiar with the formula for air resistance,
but please take a few minutes to try and make sense of all those terms that are
there!) (Don’t spend any more than 5 minutes considering these options, just get
a feel for what you think is happening.)
width of filter
height of filter
shape of filter
velocity of filter
mass of filter
material filter is made of
density of air
other?
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ACTIVITY 20. DRAG FORCE ON A COFFEE FILTER
3. In fact the book tells us that
1
Fdrag = CρAv p
2
where C is the drag coefficient, A is the cross-sectional area of the falling object, ρ
is the density of air, and p is a power that we need to determine. The book claims
that p = 2, we will verify that value. (When we learn about work and energy, we
will be able to prove this formula, if we make one reasonable assumption.)
Reconcile any major differences between your predictions on the last page and the
formula from the book.
4. Sketch the free body diagram of the filter just after it begins to fall. Be sure to
include the acceleration and velocity vectors in the diagram.
5. Write an equation which gives the acceleration in terms of the forces. Be very
careful that the signs are correct. Take the downward direction to be positive.
6. Using your last answer, write down an expression for v̇ in terms of C, ρ, A, v, g and
m.
Preliminary Document
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7. From the above differential equation, does dv/dt increase or decrease as the velocity
increases from zero? Explain.
8. Can dv/dt ever be zero? Describe the motion when dv/dt = 0.
9. Find an expression for vterminal (the velocity when dv/dt = 0) in terms of C, ρ, A, g
and m.
1. Now we that we have the general equations, we will look at a particular example of
the drag force. On the next page is the slope field where we picked the constants
so that
1 CρA
=2
2 m
and p = 2. Recall that the short lines give the slope of v at each point. What is
vterminal for this value of the parameters?
2. What is the value of the slope of v(t) when v = vterminal ?
3. Does this agree with the slope field?
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ACTIVITY 20. DRAG FORCE ON A COFFEE FILTER
4. In general, does the value of the slope depend on v? does this agree with the slope
field below?
5. In general, does the value of the slope depend on t? does this agree with the slope
field below?
6. State in words how you would sketch an approximate solution to this differential
equation if v(0) = 0 m/s. Does your solution have to hit a tangent line all the
time?
7. Sketch the approximate solutions for v(0) = 0 m/s and v(0) = 3 m/s.
slope field for a falling object
3
2.5
v(meters/second)
2
1.5
1
0.5
0
0
0.1
Preliminary Document
0.2
0.3
0.4
0.5
x(seconds)
0.6
0.7
0.8
0.9
1
University of New Hampshire
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Experiment We will now experimentally determine the power p.
Given the formulas you have just written, why can we not determine the power from
finding the terminal velocity for just one filter? Hint: what values do we know, what
values are unknown?
We can, however, find the power p by finding the terminal velocity for several different
runs, each with a different number of filters (from one to five) and therefore a different
mass. We will find the power by fitting terminal velocity vs. mass to a power law form.
1. Turn on the computer and ULI. Open up MacMotion program.
2. Make sure that the UMD is pointing straight down by making sure it tracks your
hands if they are directly underneath the UMD.
3. Repeat the following procedure five times, beginning with one filter, and going up
to five filters:
(a) If you have more than one filter, attach the filters to each other with a small
piece of masking tape.
(b) Find the mass of all the filters using the electronic balance and write it in the
table below.
(c) Drop the filter as close as possible to the UMD. The initial data will be incorrect because we’re too close to the detector, but this will allow the filter to
reach terminal velocity before it hits the floor.
(d) Practice dropping the filter so that it goes down straight without wobbling.
This step is the most important to getting good data! You may have
to take several runs to get good data. Do not include the data for five filters
if it looks as though terminal velocity was not reached. How can you tell if
terminal velocity was reached?
(e) Once you have good data, find the value of terminal velocity by highlighting
the time interval during which v was nearly constant. Use “statistics” (under
“analyze” menu) at this stage. Enter this data in the table below. (Note that
the data for zero filters you can fill in without experiment.) Estimate the error
on your terminal velocity. Explain your procedure for estimation.
(f) You can print out one good graph of v(t) for the three or four filter run.
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ACTIVITY 20. DRAG FORCE ON A COFFEE FILTER
number of filters
mass (gm) terminal velocity (m/s) error in velocity (m/s)
1
2
3
4
5
Data Analysis
1. Open up the program Graphical Analysis on the computer.
2. Enter the values of the masses under X data, the velocity under Y data.
3. Now, let’s think about where we’re headed with the program. Below, write the
formula for the terminal velocity in terms of C, ρ, A, v, g and m.
4. You will be fitting your data to a power law y = AxB . Relate each of the general
variables in the fitting formula (y, x, A, B) to the variables in the original equation
(vt , C, ρ, p, A, m, g):
• y=
• x=
• A=
• B=
5. From the answer to your last question, how will fitting the data to a power law
help you to determine p?
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6. Let’s proceed with the fitting. In order to have a clear graph, do the following:
(a) By clicking on Label and Unit portions of the table, give each set of data the
correct label and units.
(b) Under Graph, choose Rename to give your plot a meaningful name.
7. Under Graph and Graph Options... (last item) enter a reasonable error for all
the velocity data points. (We would like to enter a different error for each data
point, but the program cannot do that.) Be sure to uncheck the percentage error
button. The under Graph, select Error Bars so that the error bars show.
8. Under Analysis choose Automatic Curve Fit and then Power - the program will
find the best power law fit for the data. Print out the graph out if you’d like.
Conclusions
1. Based on your data and the fit by graphical analysis, what is the most reasonable
value for p? How sure are you of your value of p? Is this in agreement with the
text? (Note that you can choose ”manual data fit” under the ”analyze” menu and
manually change the value of B to see what other values seem to fit the data.)
2. Based on your value of A the parameter from the fit, what is the value of C=
drag coefficient? Note that your answer should be around one. (You will have to
estimate the cross-sectional area of the filter.)
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Preliminary Document
ACTIVITY 20. DRAG FORCE ON A COFFEE FILTER
University of New Hampshire
Part III
Conservation Laws
87
Activity 21
Balance Point
In the following set of exercises we will learn how to find the balance point of an object.
The balance point is the place where you can place your finger and be able to support
the entire object. We will see in the next class that this point is also essential for
understanding motion of the object.
1. Finding the Balance Point Experimentally
Obtain a meter stick, two weight hangers and a set of weights. Place the weight
hangers at the specified locations (measured from the center of the meter stick) and
find the mass m1 needed to make the center of the stick (here taken to be 0 cm)
the balance point of the system. Note that the hangers themselves have a mass of
about 20 kg; this must be included in m1 and m2 . Finally, the smallest object we
have has a mass of 10gm, so you will not be able to get the masses correct to better
than 10 gm.
x1
m1
x2
m2
-20 cm
20 cm 70 gm
-20 cm
40 cm 20 gm
-20 cm
15 cm 40 gm
2. Verifying the formula for the balance point
The book gives the following definition for finding the balance point:
xbalance point ≡
1
N
X
Mtotal
i=1
mi xi
where N is the total number of objects that we are considering.
In the following space verify that this formula gives the center of mass at the center
of meter stick for each of the three situations given above. (Ignore the mass of the
meter stick itself in this calculation.)
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ACTIVITY 21. BALANCE POINT
3. No-calculation balance points
In following three cases, imagine that the object is a thin sheet of metal of the
specified shape and uniform density, and that you are trying to balance it with
your finger while the object is horizontal.
Mark the balance point of a meter stick with no extra hanging weights.
Mark the balance point of a square book.
Mark the balance point of a circle.
How did you determine these balance points?
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4. Calculating Balance points in two dimensions
Now imagine that you have a square plate on which you place three objects with
the following locations and masses:
x
y
m
0 m 0 m 3 kg
1 m 2 m 4 kg
2 m 1 m 8 kg
Sketch the location of the objects on the coordinate system of the plate and use
your intuition to guess the location of the balance point. Mark your guess on the
sketch with a ”g”. (Ignore the mass of the plate itself in this calculation.)
2
y
1
2 meters
1
x
Using the above formula for x balance point and
ybalance point ≡
1
N
X
Mtotal
i=1
mi yi
calculate the balance point and mark that on your sketch with an ’x’.
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ACTIVITY 21. BALANCE POINT
5. Calculating the Balance point in pieces
To illustrate a useful technique we will calculate the balance point in a different way.
First calculate the balance point of the first two objects only in the last section:
Now, consider those first two objects to be one object with a mass of 7 kg located
at their balance point. Now calculate the balance point for the ”7 kg object” and
the 8 kg object.
You should find that the balance point calculated in this section and the last are
the same. If not, go back and check your calculations.
For those of you that like proofs, can you show that these balance points must be
the same?
6. Calculating Balance points for complex objects
Use the ideas developed in this sheet to calculate the balance point for the following
situation. Be sure to indicate what you are doing and why. The mass of the
rectangle is 5kg, the mass of the circle is 3kg.
2
y
1
2 meters
1
x
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7. Calculating the Balance point for even more complex objects
Imagine that you have a rectangular sheet of length l = 1 m and width w = 1/7 m
constructed so that the density is
density = 3x2 = mass/area
Where x is the distance along the sheet, with x = 0 at the left side and 3 is in units
of kg/m4 . With this density, the sheet is much heavier on the right end than on
the left. Sketch the density vs. x below. Calculate the values at each end of the
bar.
Mark with a ”g” where you would guess the balance point is located.
What is the value of ybalance point ? Explain.
Calculating xbalance point is more difficult and will take several steps. Why do none
of our other methods that we have used so far work here?
P
In this case we will need to approximate the total mass ( mi ) and first moment
P
of the mass (defined as mi xi , the numerator in the center of mass formula) by
breaking the sheet into pieces and approximating the density of each piece to be
constant. (This is consistent with what we did in the last exercise: calculate the
center of mass in pieces.) Do you expect this approximation to be accurate for two
pieces? four pieces? one thousand pieces? Explain.
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ACTIVITY 21. BALANCE POINT
(a) To calculate xbalance point , first imagine that the mass of the sheet is located at
only two points as illustrated below.
Fill out the chart below, keeping numbers as fractions, keep the powers explicit
(e.g. you should have terms like (1/4)2 , not 12 /8). The reason for this is that
we are looking for a general pattern which will not be obvious otherwise.
piece 1
piece 2
∆x
size of piece
x
location of dot
density
at the dot
area
of the pieces
mass
assuming constant density
mx
first moment of the mass
(b) Do the same thing as in the previous question, considering that all the mass
is located at four evenly spaced points along the sheet.
piece 1
piece 2
piece 3
piece 4
∆x
size of piece
x
location of dot
density
at the dot
area
of the pieces
mass
assuming constant density
mx
first moment of the mass
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P
(c) We have calculated the terms for the total mass mi and first moment
Rewrite the terms in this sum using the density instead of the mass.
P
mi xi .
(d) Based on your answers to the last two questions, come up with a general form
for the terms of the i’th piece if you had n points.
piece i (0 ≤ i ≤ n)
∆x
x
density
area
mass
mx
(e) Next you will use Matlab to calculate xbalance point for several different values
of n. Note that you can either take the points to be at the left edge (0 ≤ i ≤
(n − 1)) or at the right edge (1 ≤ i ≤ n). Don’t forget to use the up arrow key
to re-run the code for different values of n. Also, recall that sum(f(2:N+1))
sums the second through N+1 elements in the vector f.
Write out your matlab code below:
Write down the values obtained below:
n
total mass (left) total mass (right) mx (left) mx (right)
2
5
10
100
1000
(f) Why do the values of total mass and balance point change as you change the
number of points? Why do they change less as the number of points gets
larger?
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ACTIVITY 21. BALANCE POINT
8. Balance point of the bird
An unusual bird will be passed around the class. Where do you think the balance
point of the bird is? Hint: what other object behaves like the bird? Where is the
center of mass of that object in relation to its support?
Why does the bird behave as it does?
9. Balance point of the bottle
Sketch the wine bottle and holder below. Where is the balance point? You can
estimate the balance point from geometry and from looking at the wooden stand.
Do those values agree?
10. Estimate of balance point
Below is the sketch of another bird made out of a piece of wood of uniform density.
Which of the numbered points is most likely to be the balance point? Hint: first
elminate those that are obviously incorrect. For the rest, draw axes through the
points, compare mass distribution above and below and left and right.
1
5
4
2
6
3
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University of New Hampshire
Activity 22
Proof of Conservation of Momentum
Consider a complex system made of many particles. A good example is the system
of three blocks being pushed from the outside over a rough surface with coefficient of
friction µk . In any system there are both internal forces (forces that the blocks exert
on each other) and external forces (forces exerted by bodies outside the system, such
as friction, push, and the gravitational pull of the earth).
3M
M
2M
Fhand
B
C
A
frictionless table
To help you understand the proof below, sketch the free body diagram for each of the
blocks and label each as follows
1. the object on which the force acts (second subscript),
2. the object exerting the force (first subscript), and
3. the type of force (N=normal, f=frictional, G=gravitational, T=tension etc.)
Below are all of the mathematical steps to prove an important law of physics for
complex systems. Your job is to provide a justification (for example, definition, law of
physics, algebra, calculus) for each of the following steps in the proof.
~xcm = M1tot (m1~x1 + m2~x2 + m3~x3 )
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ACTIVITY 22. PROOF OF CONSERVATION OF MOMENTUM
Mtot~xcm = m1~x1 + m2~x2 + m3~x3
Mtot~vcm = m1~v1 + m2~v2 + m3~v3
Mtot~acm = m1~a1 + m2~a2 + m3~a3
Mtot~acm =
P~
P
P
F1 + F~2 + F~3
Mtot~acm = F~1 , net + F~2 , net + F~3 , net
Mtot~acm = F~1 , net, external + F~2 , net, external + F~3 , net, external
+F~1 , net, internal + F~2 , net, internal + F~3 , net, internal
Mtot~acm = F~1 , net, external + F~2 , net, external + F~3 , net, external
Mtot~acm =
P~
Fnet,external
The formula on the last page is Newton’s second law for a system of particles. It
always holds true (at least in this class!). There is one special case worth noting. If
P~
Fexternal = 0, what can you say about Mtot~vcm ? Explain.
Preliminary Document
University of New Hampshire
Activity 23
Recognizing Conservation of
Momentum
In each of the following scenarios you are given a system of objects. Draw the free body
diagram of each object in the system, and decide if momentum is conserved for the entire
system by looking at the net external force on the system.
1. Consider a person walking on a boat in still water. The system is the boat and the
person.
2. Consider a Mack truck colliding with a VW bug on rough pavement. The system
is the truck and the VW bug.
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ACTIVITY 23. RECOGNIZING CONSERVATION OF MOMENTUM
3. Consider a bullet being fired into a block of wood. The wood is resting on a
frictionless surface and begins to move as soon as the bullet hits it. The system is
the bullet and the block.
4. Consider a bullet being fired into a block of wood. The wood is suspended from a
rope and begins to swing upward as soon as the bullet hits it. The system is the
bullet and the block.
5. Consider an exploding coconut resting on ice. When the coconut explodes, the
pieces fly off in many directions, but all the pieces remain in contact with the ice.
The system is just the coconut.
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Activity 24
Air Drag and Euler’s method
Overview: In this worksheet you will use Euler’s method to find the solution to the
differential equation for velocity for a falling coffee filter. You will compare your approximate solution with the exact solution, which we have seen, but don’t yet know how to
derive.
Deriving the differential equation: Consider again an falling obejct subject to
both gravitational force downward and air drag upward. You know from Newton’s laws
that
m a = −Fdrag + mg
where we’ve taken the downward direction to be negative. From the text book,
1
Fdrag = CρAv 2
2
where C depends on the shape, (take C = 1), ρ is the density of air, and A is the
cross-sectional area of the object.
From the statements above, show that
v̇ = −dv 2 + g
where d = 12 CρA/m.
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ACTIVITY 24. AIR DRAG AND EULER’S METHOD
Understanding the Exact Solution: Next semester we will derive the solution to
this differential equation (with v(0) = 0 m/s) is as follows:
√
√
g(−1 + e2 gd t )
√
v(t) = √
d(1 + e2 gd t )
For now, we will have to accept it as true. But we can at least check that it is
reasonable:
• Given this formula, what is v(0)?
• What value does v(t) approach when t is large? Hint: Are the exponential terms
large or small (compared to one) as t gets very large?
• What is the terminal velocity (when v̇ = 0)? How does that compare with your
last answer?
Sketch this solution on Matlab for d = 2 for the time interval 0 ≤ t ≤ .8 seconds.
(Yes, Matlab! - we will need the ”for” loop later, so we might as well fire it up!) Sketch
it here.
Does the plot of v(t) look reasonable? Explain.
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Estimate the solution using Euler’s method Now use Euler’s method to solve
this equation. Recall that the general form of Euler’s method is, given that
y 0 (t) = f (x, y)
then
yn = yn−1 + ∆xf (xn−1 , yn−1 )
• Write out Euler’s method for this problem. That is, what variable plays the role of
y? x? what is f (x, y)?
• What is the initial value, y0 ?
• Write the Matlab code here:
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ACTIVITY 24. AIR DRAG AND EULER’S METHOD
• Use Euler’s method to solve this differential equation for 0 ≤ t ≤ .8 seconds, using 10 steps. Plot the results on top of the exact solution by using the command
plot(t,v,’*’,t,vexact); this way the approximate solution is show as ”*”, and
the exact solution is shown as a line. Does your approximate solution seem reasonable? Explain.
• Use Euler’s method to solve this differential equation for 0 ≤ t ≤ .8 seconds,
using 20 steps. Plot your results as before. How does this solution compare to the
previous one?
• Use Euler’s method to solve this differential equation for 0 ≤ t ≤ .8 seconds, using
100 steps. Plot your results as before. How does this solution compare to the
previous one?
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University of New Hampshire
Activity 25
Forces and work
Your group will work with the following force:
For your force, answer the following questions:
• How do we determine the magnitude of this force; i.e., is there a general formula?
• Can this force do positive work, that is, can it speed something up? If so, give an
example
• Can this force do negative work, that is, can it slow something down? If so, give
an example
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ACTIVITY 25. FORCES AND WORK
• Can this force do no work as it acts on a moving object? If so, give an example
• Can you write a general formula for the work done by this force? If so, write it
down. If not, explain why not.
• Can you come up with a situation in which this force does work on a round-trip
path (that is, a path that begins and ends at the same point)? One easy way to do
this is to think of a situation in which your force is the only force doing work. If
the kinetic energy changes on a round trip path, then by the work-energy theorem,
the net work done is NOT zero.
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University of New Hampshire
Activity 26
Proof of Work-Energy Theorem
As we did with conservation of momentum, we will prove the work-energy theorem. The
tutorial showed it was true in two cases with constant force. We can prove it is true in
general. Below are the steps in the proof; please provide justification for each step (for
example, defnintion, algebra, calculus).
Note, we use the notation d~s = dx î + dy ĵ for a small spatial displacement. Also note
~·B
~ = ax bx + ay by = |A||B| cos θAB .
that the dot product can be written: A
R
Work = W = F~ · d~s
R
W = (Fx dx + Fy dy)
R
W = (max dx + may dy)
R
W = (m dvdtx dx + m dvdty dy)
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ACTIVITY 26. PROOF OF WORK-ENERGY THEOREM
R
W = (m dvdtx dx
dt + m dvdty dy
dt)
dt
dt
R
W = (m dvdtx vx dt + m dvdty vy dt)
R
W = (mvx dvdtx dt + mvy dvdty dt)
R
W = (mvx dvx + mvy dvy )
W =
³
1
mvx2
2
´
v
+ 12 mvy2 |vfi
v
W = 12 mv 2 |vfi
W = KE|final
initial
W = KEf − KEi = ∆KE
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University of New Hampshire
Activity 27
Proof of Conservation Laws
In this activity you will proof both the conservation of mechanical energy and then the
conservation of linear momentum. Neither is conserved always. You will need to find out
what you have to assume in order for each quantity to be conserved
Conservation of Mechanical Energy
Below we will prove together the conservation of mechanical energy. We give the first
and last steps, you need to fill in the in between steps - there are about three steps. You
can work backward if you that seems easier!
Unlike the Work-Energy Theorem which is always true, you will find that you need
to make one stipulation in order for mechancial energy to be conserved.
∆KE = W
∆KE + ∆P E = 0
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ACTIVITY 27. PROOF OF CONSERVATION LAWS
Proof of Conservation of Linear Momentum
Below we will prove together the conservation of linear momentum. We give the first
and last steps, you need to fill in the in between steps - there are about two steps. You
can work backward if you that seems easier!
Unlike the Newton’s Second Law which is always true, you will find that you need to
make one stipulation in order for linear momentum to be conserved.
d~
pcm
dt
= F~net, external
p~cm = constant
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Activity 28
Contrast Work-Energy Theorem and
Fnet = ma
In order to understand the differences between the Work-Energy Theorem and Newton’s
second law (Fnet = ma), you need to use both methods on the same problem. Below are
three problems where you are to do just that. You will find that not every question that
we ask can be answered, but you should be clear about why they can’t be answered.
A 5 kg block rests on a 30 degree incline a distance of 1.3
• m up the ramp. The ramp
is frictionless. The block is
released from rest.
d=1.3m
– Use the Work-Energy Theorem to find the velocity at the bottom of the ramp.
– Use the Work-Energy Theorem to find how long it takes to get to the bottom
of the ramp.
– Use Newton’s second law to find the velocity at the bottom of the ramp.
– Use Newton’s second law to find how long it takes to get to the bottom of the
ramp.
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ACTIVITY 28. CONTRAST WORK-ENERGY THEOREM AND FNET = M A
A 5 kg block rests on a
curved ramp, a horizontal
distance of 1.3 m above the
•
ground. The ramp is frictionless. The block is released from rest.
v=3m/s
h=1.3m
– Use the Work-Energy Theorem to find the velocity at the bottom of the ramp.
– Use the Work-Energy Theorem to find how long it takes to get to the bottom
of the ramp.
– Use Newton’s second law to find the velocity at the bottom of the ramp.
– Use Newton’s second law to find how long it takes to get to the bottom of the
ramp.
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Consider a block on a rough surface.
It is held against a spring (but not
attached to the spring) so that the
spring is compressed leftward 3 cm.
It is then released from rest, and
the block travels rightward, separates from the spring after traveling
•
3 cm, and then eventually comes to
rest after traveling a total distance
d from where it started. The block
has a weight of 5 kg (take g = 10
m/s), the coefficient of kinetic friction is µk = .13, and the spring constant is k = 13N/cm.
at rest
x=-3cm
d
x=0=x eq
at rest
– Use the Work-Energy Theorem to find the distance d.
– Use the Work-Energy Theorem to find how long it takes to come to rest.
– Use Newton’s second law to find the distance d.
– Use Newton’s second law to find how long it takes to come to rest.
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ACTIVITY 28. CONTRAST WORK-ENERGY THEOREM AND FNET = M A
Consider a pendulum of length L = 1.2 m with a small
ball of mass 32 gm hanging on the end. The ball is released from rest when the string makes a 23o angle with
the vertical.
23
L
– Use the Work-Energy Theorem to find the velocity at the bottom of the swing.
– Use the Work-Energy Theorem to find how long it takes to get to the bottom
of the swing.
– Use Newton’s second law to find the velocity at the bottom of the swing.
– Use Newton’s second law to find the velocity at the bottom of the swing.
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Activity 29
Conservation Lab I
Note: This will be handed in! Please write neatly and clearly.
In the first part of this lab, you will push a cart loaded with four weights toward
another unloaded stationary cart. The carts will be velcro sides together and so will
stick on impact. You will take data with the motion detector.
1. Is any quantity conserved?? We need to find out if mechanical energy or linear
momentum or both are conserved for the system of the two carts.
• Draw the free body diagrams of both carts during the collision
• Given your free body diagrams, is mechanical energy conserved during the
collision? Explain.
• Given your free body diagrams, is linear momentum conserved during the
collision? Explain.
2. Check with an instructor before continuing.
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ACTIVITY 29. CONSERVATION LAB I
3. Making predicitons At this point you should see that linear momentum is nearly
conserved (if we ignore friction). We will come back to friction later!
Before we move on, it is important to be careful about notation. The usual notation
for velocities before are v1i and v2i , and for the velocities after we write v1f and
v2f . There are four velocities here and it is essential that they don’t get confused!
In our experiment v2i = 0 and v1f = v2f ≡ vf .
Use conservation of linear momentum to write down the velocity of the two carts
after the collision (vf ) in terms of the velocity of the loaded cart before the collision
(v1i ) and the masses of the two carts.
4. Plan the experiment What data must you take in order to verify conservation
of momentum in this setup?
5. Prepare to take data Here is some advice on getting good data:
• Clean and level the track.
• Be sure the beam of the motion detector is aimed at the carts. You can do this
by doing a test run and verifying that the computer sees the whole motion.
• Use 15 point averaging to obtain smooth data. Use 40 points per second data
rate to get accurate times for the collision.
• Send the loaded cart gently toward the stationary cart. A hard collision may
introduce vibrations.
• You may need to take data several times until you get clean data.
• Check with an instructor on the quality of your data before proceeding.
• One accurate way of reading data off of the computer is to open up the data
table and look at the values there. In this way you don’t need to rely on the
accuracy of your hand and the mouse.
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6. Taking and Analyzing the Data
• Do several runs until you get one that is reasonably unnoisy. Print out the
velocity plot.
• Look at your plots of v(t). There is a slight downward slope to that line.
Why?
• We will deal with the downward slope in stages. First this means that velocity
is not constant before or after the collision. Look now at the a(t) plot (narrow
the time range so you’re just looking at the time around the collision). Looking
at this plot, how can you identify the time just before the collision and the
time just after?
• What is velocity just before the collision? just after?
• What are the masses of the carts?
• Using the above data, calculate the linear momentum before and after the
collison.
• What is the percentage difference between the two? Percent difference =
(pfinal − pinitial )/pinitial t.
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ACTIVITY 29. CONSERVATION LAB I
• Is the final or initial momentum greater? Why should this be so?
• Now we will try to correct for fricton. Using the impulse momentum theorem,
how do we calculate the change in momentum due to frictional force?
• Obtain a reasonable value for the average acceleration due to friction from the
acceleration plot.
• Write down the ∆t for the collision from the previous page.
• Plug these values and the momentum values in your formula to see if there is
now better agreement between your prediction and the data.
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7. More kinds of Collisions There are three general categories of collisions: elastic,
inelastic, and completely inelastic.
• In elastic collisions, mechanical energy is conserved as well as linear momentum.
• In completely inelastic collisions the objects stick together afterwords and
mechanical energy is lost to heat and sound. We just did a completely inelastic
collision.
• Inelastic collisions have some energy loss, but the objects do not stick together.
This is the most ususal situation, but we cannot take data for such a collision
since we need to get two velocities after collision.
8. Extra Credit Elastic Collision We can also do an experiment with an elastic
collsion. In this case we use the magnetic carts and send one cart at another
stationary cart.
The book (see page 203) shows that if v2i = 0 (that is, the second cart is initially
at rest), then
v1f =
m1 − m2
v1i
m1 + m2
v2f =
2m1
v1i
m1 + m2
and
(These equations come from conservation of mechanical energy and linear momentum; the algebra is a bit difficult, so we do not do it here!)
If you have time, you can also verify conservation of mechanical energy and linear
momentum in this elastic collision.
Have two equal mass carts. Initially only one is moving and the other is still. Send
the moving one gently at the stationary one. Be sure they have magnetic ends
together so that the carts don’t actually touch during the collision.
• Theory How do the magnets allow mechanical energy to be conserverd?
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ACTIVITY 29. CONSERVATION LAB I
• Given that we have equal mass carts, what should be v1f - the velocity of the
initially moving cart after the collision?
• In terms of v1i , what should be v2f - the velocity of the initially stationary
cart after the collision?
• Data What data do you need to take for this experiment?
• Taking data Go back and review the hints on taking good data. Take your
data and write the key values below. Print a plot of the velocity data.
• Is linear momentum conserved in this experiment? Explain.
• How important is friction in this experiment?
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University of New Hampshire
Activity 30
Conservation Lab II
Note: This will be handed in! Please write neatly and clearly.
In this lab you will investigate a pendulum: a small object hanging on a long string.
1. Is any quantity conserved? We need to find out if mechanical energy or linear
momentum or both are conserved for the swinging object.
• Draw the free body diagrams of the swinging object.
• Given your free body diagrams, is mechanical energy conserved during the
swing? Explain.
• Given your free body diagrams, is linear momentum conserved during the
swing? Explain.
• Check with an instructor before continuing.
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ACTIVITY 30. CONSERVATION LAB II
2. Making predicitons At this point you should see that mechanical energy is nearly
conserved (if we ignore air friction).
• Sketch a picture of the pendulum at both top and bottom of the swing (not a
free body diagram, but a full diagram). Label all the important parameters.
• What is the mechanical energy at the top of the swing in terms of your parameters?
• What is the mechanical energy at the bottom of the swing in terms of your
parameters?
• What values will you need to measure to check that these values are the same?
• How will you measure those values? Note that some values will have to be
measured indirectly.
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3. Prepare to take data Here is some advice on getting good data:
• Make the string quite long (about a meter). Use the yellow fly fishing backing
because it knots easily and doesn’t stretch.
• Use at least a 200 gram hanging mass. It needs to be at least that big to
create a good echo for the motion detector.
• Let the swing of the pendulum be quite small - about 2 cm, otherwise it will
swing in and out of the beam of the motion detector.
• Take several minutes to be sure that the pendulum is in the path of the motion
detector. Wave your hand back and forth to determine the beam size. Adjust
either the pendulum or the detector so the pendulum is in the beam.
• Be sure that the pendulum support is not in the beam. You can also wrap
something fuzzy around the rod so it will not produce a good echo.
• Use 15 point averaging to obtain smooth data. Use 50 points per second data
rate to get accurate data.
• You may need to take data several times until you get clean data. Often
some of your data will be of poor quality, but if you have four or five good
oscillations, that is all you need!
• To improve the accuracy of your data, take data for several different oscillations and average.
• Check with an instructor on the quality of your data before proceeding.
• One accurate way of reading data off of the computer is to open up the data
table and look at the values there. In this way you don’t need to rely on the
accuracy of your hand and the mouse.
4. Taking and Analyzing the Data
• Do several runs until you get one that is reasonably unnoisy. Print out the
distance plot.
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ACTIVITY 30. CONSERVATION LAB II
• Write down the values of the parameters that are constant for your pendulum:
• Decide what values you will need to calculate the mechanical energy at top
and bottom of the swing. How will you get each of those values from the data?
• On the chart on the next page, label each column for data you either get off
the computer or which you need to calculate.
• Check with an instructor before proceeding.
Preliminary Document
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125
Column 1 Column2 Column 3 Column 4 Column 5 Column 6 Column 7
time
seconds
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ACTIVITY 30. CONSERVATION LAB II
• Is mechanical energy conserved? Explain.
5. Extra Credit - what about friction?
If you have time, try to estimate the energy lost to friction. Explain your method
and show your data below.
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Part IV
Rotational Motion
127
Activity 31
Rotational analogs to force and mass
In the following exercises we will investigate the rotational analogs to force and mass,
and see that things are a bit different for the rotational situation.
1. Investigating the analog to force Obtain two meter sticks, two weight hangers
and a set of cylinder weights. Use the 100 or 200 g cylinders for this investigation.
(a) Take one meter stick. Place the weight at 30 cm and hold it at the 0 cm end.
Have one of your partners slide the mass slowly out to 1 m, trying not to
support the mass as they slide it. How does your effort change, if at all, as
the mass slides out to the end?
(b) Place the weight at 80 cm, but hold it so that the angle between the vertical
and the stick is 90 degrees and then 45 degrees and then 0 degrees (all of these
angles will be approximate, of course). Rank these in the order of difficulty
to hold, most difficult to least.
cylinder at 45
degree angle
(c) Make a door rotate, first by pushing at the edge of the door, then at the inner
edge (the side with the hinges). Use your knuckle to do the pushing so that
you have to push rather hard in both cases. Which is harder to move?
(d) Look back on your investigations. What does the effort you were required to
make depend on? Just the force (the weight of the hanging object) or were
there other factors? Explain.
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ACTIVITY 31. ROTATIONAL ANALOGS TO FORCE AND MASS
(e) The book defines something known as torque
τ = rF sin θrF
where r is the distance between the pivot point and where the force is applied,
F is the force and θrF is the angle between ~r and F~ . Lets assume torque is a
measure of your effort. Does this equation agree with what you found in your
experiments? Explain.
2. Investigating the analog to mass
Consider two meter sticks with attached cylinders of 100 g; one cylinder is attached
at 10 cm on stick A, the other at 90 cm on stick B.
(a) Predict which one will be harder to balance. Predict which one will hit the
ground first if allowed to fall from rest. Explain your reasoning.
(b) Now try balancing. In this case you should not hang the cylinders from
the usual position, but hang them so they are on the wide side of the stick.
This requires hanging them from the “wrong” place. Which one is harder to
balance?
(c) Now allow them to fall. Hold both sticks still, in vertical position, with a third
stick placed horizontally near their top, then release them at the same time.
As in the last experiment, hang the weights on the wide side of each stick.
Which one fell first?
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In the following steps we will try to explain the results of the last experiment
and see that we run into trouble.
(d) Which stick had the greater torque?
(e) Did the stick with the greater torque fall faster?
(f) How do the masses of the two systems A and B (system = stick plus cylinder)
compare? That is which is bigger, or are they the same?
(g) Let’s consider a one-dimensional analogous system: two objects being accelerated in one-dimension. They both have the same mass, and the same initial
velocity of zero. If object A is acted on by a greater force than object B,
which will move faster?
(h) Did the linear analogy help explain which fell first? Explain.
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ACTIVITY 31. ROTATIONAL ANALOGS TO FORCE AND MASS
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University of New Hampshire
Activity 32
Moment of inertia
First, recall that we found in the last activity that mass and the distance of that mass
from the pivot both effect how difficult an object is to rotate. Moment of inertial is
therefore the rotational analog to mass.
In this exercise we will calculate the moment of inertia for several objects, both point
particles and extended objects. the moment of inertia is defined as
X
IA ≡
2
mi rAi
over all particles
is the moment of inertia of a set of particles about axis A, and rAi is the distance between
the axis A and the ith particle. In this worksheet we will use this formula to calculate
the moment of inertia of several different objects.
1. Point particles We begin with the easiest situation. Consider a system with two
point particles: one of mass 3 kg at x = 0 cm, and another of mass 5 kg at x = 2cm.
• What is the moment of inertia of this system of particles for an axis through
the origin?
• for an axis through x = 2 cm?
• for an axis through the center of mass?
• for an axis through x = 100 cm?
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ACTIVITY 32. MOMENT OF INERTIA
2. Intuition Given your answers above, which do you suppose would have a greater
moment of inertia through its center: a sphere of mass M and radius R or a hoop
of the same mass and radius? (A hoop has essentially all of its mass at radius R)
Explain.
sphere
hoop
3. Center of Mass One very useful theorem for calculating the center of mass is the
parallel axis theorem which is stated as follows:
IA = Icm + M h2A
where hA is the distance between the center of mass and the axis A, and Icm is the
moment of inertia when the axis goes through the center of mass. The two axes
(axis A and the axis through the center of mass) must be parallel. (The proof of
this is in section 11.7 of the book.)
• Given the theorem above, how can we see that the moment of inertia for any
object is smallest through the center of mass?
• Was the moment of inertia smallest through the center of mass in problem
one?
• Why is it reasonable (use your intuition) that the moment of inertia for any
object is smallest through the center of mass?
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4. Moment of inertia for a hoop - you have all the tools you need to solve these
two problems below. Go back and see what equations seem useful.
• Find the moment of inertia of a hoop for an axis through its center if it has
a mass M and radius R. (Axis A in the picture.) Recall that a hoop has
essentially all of its mass at radius R.
B
A
• Find the moment of inertia of a hoop for an axis through its edge. (Axis B in
the picture.)
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ACTIVITY 32. MOMENT OF INERTIA
5. Moment of inertia for a rod of uniform density In this section we will find
the moment of inertia of a rod of uniform density for an axis through its center.
Take the rod to be of length L and mass M . We will calculate the moment of
inertia in the steps outlined below.
(a) Why will we have to use calculus here?
M
Axis of rotation
L
(b) Choose an origin to measure distances from. Mark it on the diagram. Be sure
to measure all distances with respect to this origin.
(c) Break up the rod into lots of little chunks and find the mass of each chunk.
Hint: what is the linear density (mass/length) of the whole rod?
(d) What is the contribution to I for each chunk? Write total I as a Riemann
sum. Hint: you may find it helpful to distinguish between parameters (e.g. M
and L) that don’t change for a given object and variables (e.g. x or r) that
change even for a given object.
(e) Take the size of the chunk to zero so that the sum becomes and integral. What
are the limits on the integral?
(f) Evaluate the integral.
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(g) Does your answer for I of the rod make sense? For example, should I for
the rod be more or less than if the mass were concentrated in two spheres a
distance L apart?
(h) Now use the same procedure to calculate I for an axis through one end of the
rod.
Axis of rotation
M
L
(i) Check your last two answers using the parallel axis theorem.
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ACTIVITY 32. MOMENT OF INERTIA
6. Moment of inertia for a cylinder for an axis through its center Take the
total mass to be M , height to be h, and the radius to be R. Again here, you will
have to use calculus.
• As you break the cylinder into chunks, remember that each chunk must have
a single value of r - the distance to the axis. How will you chunk the cylinder
given this constraint?
• What is the density (mass per volume) of this cylinder?
• What is the mass in a chunk?
• What is the contribution to I for each chunk?
• What is the Riemann sum for I? What is the corresponding integral and
limits?
• Evaluate the integral.
• Note that we can’t do all shapes because it requires multi-dimensional calculus.
The table on page 249 will help on homework and will be provided on tests if
needed.
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University of New Hampshire
Activity 33
Spinning our selves!
1. In this activity we will investigate rotations
(a) We begin with some investigations in a rotating chair. Caution! if you
become dizzy easily, you probably should not do this investigation! In the
experiments below, be sure that everyone who wants a turn on the chair has
a turn on the chair.
• Have one person sit in the chair and hold the bicycle wheel horizontally.
Have the seated person spin up the wheel. What happens to the person
and chair?
• Explain what happen in terms of torque. Are there torques exerted on
the wheel? on the person? who/what exerts those torques?
• Now have the seated person stop the wheel with their hands. Explain
what happens to the person and why (in terms of torque).
• Now have a second person hand the seated person an already spinning
wheel. Have the seated person turn the wheel over. Explain what happens
to the person and why (in terms of torque).
• Have the seated person hold a ball in one hand, extended away from the
body and the throw the ball straight. Explain what happens to the person
and why (in terms of torque).
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ACTIVITY 33. SPINNING OUR SELVES!
• Consider the equation
X
~τ =
~
dL
dt
~ change?
If the system is isolated and has no external torque, how does L
• Use your answer above to explain what happened in at least one of the
chair exercises. Be sure to identify your system.
• Lastly, have a person sit in the chair holding weights in their hand, with
hands in close to their chest. Have another person spin them up in the
chair. Then have the person extend their hands (with the weights) outward. Explain what happens in terms of conservation of angular momentum.
• Predict what will happen if the seated person
i. bring their hands straight up
ii. extends both their hands and feet
then check your predictions.
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(b) The second part of this investigation is to do a few problems related to the
conservation of angular momentum:
i. Consider a student seated on a stool that can rotate freely about a vertical
axis. The student, initially at rest, is holding a bicycle wheel which is also
at rest. The moment of inertia of the wheel is 1.2 kg m 2 , the moment of
inertia of the student and stool is 5.6 kg m 2 . The student then spins up
the wheel so that it is rotating with an angular speed of 3.9 rad/sec. How
fast is the student/stool system rotating after the wheel is spun up?
ii. Consider the same student who now is initially at rest and is holding the
same wheel with angular speed of 8.3 rad/sec counterclockwise. She then
turns the wheel over so that its angular speed is now 8.3 rad/sec clockwise.
What is her angular speed after she turns the wheel over?
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ACTIVITY 33. SPINNING OUR SELVES!
iii. Consider the same student sitting in the same stool. In an outstretched
arm, extended .7 m from her center, she catches a baseball of mass 200
g moving at 30 m/s directly into her hand. What is her angular velocity
after she catches the ball? Her moment of inertia with an outstretched
arm is 5.7 kg m2 .
iv. A bug of mass 1 g is on a rotating record of mass 10g and radius 8 cm.
When the bug is at r = 4 cm, the record and bug rotate at an angular
velocity of 2 rad/sec. If the bug walks out to 7.5 cm, how fast will the
system rotate?
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University of New Hampshire
Activity 34
Moment of Inertia Experiment
Next week you will have one class period to measure the moment of inertia of one the
large metal pulleys used in the lab.
1. How would you calculate the moment of inertia of the pulley just using a ruler
or other device for measuring length? (It is made of aluminum, with density2.7
g/cm3 .) Don’t do the work here, just describe it.
2. How would you calculate the moment of inertia of the pulley using a motion detector, carts, masses, strings, or any of the other equipment in the lab? Again, don’t
do the work yet, just describe it. Hint: think about the problems we did in the last
physics class with objects attached to pulleys by strings.
3. Next week we will ask you to carry out your plan to calculate Ipulley using the
motion detector etc. and using the dimensions of the pulley (these are two separate
methods). The last half hour of class you will need to spend writing up what you
did. The write-up must include:
• A clear description of your method.
• The data that you used (including plots and tables as necessary).
• The calculuations you did to calculuate the moment of inertia.
• A short summary. Did you get reasonable agreement between the two methods?
Note that you can have someone write up these things as you go, so that you have
more time to do the experiment. These reports will be handed in.
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ACTIVITY 34. MOMENT OF INERTIA EXPERIMENT
University of New Hampshire
Activity 35
Problem Using Torque and Moment
of Inertia
Consider an Atwood machine for which the pulley has a mass of 2.5 kg and radius of .2
m. On one side of the pulley hangs a block with mA = 1.2 kg, and on the other side
hangs a block with mB = .5 kg. What is the linear acceleration of the system?
(Hint: the tension on one side of the pulley must not be the same as the tension on
the other side. What would the net torque on the pulley be if the tensions were the
same? The tensions were the same before because we assumed a massless pulley)
A
B
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ACTIVITY 35. PROBLEM USING TORQUE AND MOMENT OF INERTIA
Preliminary Document
University of New Hampshire
Activity 36
Atwood Machine Problem - Redux
We reconsider a problem that we have done before, this time allowing the pulley to have
a non-negligible mass.
In the figure below there is a cart with mass M which moves without friction over
the table top. It is attached to a cord that wraps over a frictionless pulley to a second
block with mass m. The cord is massless and the pulley has a moment of inertia I. The
block falls and accelerates the cart to the right.
What is the acceleration of the hanging block? What is the tension in the horizontal
cord? What is the tension in the vertical cord? (Why are the tensions not the same?)
A
B
To gain a better understanding of this typical problem, do the problem in several
ways (each is more difficult than the last)
1. Take µ = 0 (no friction) and use numerical values: M = 5kg, m = 8kg, I = 2kg
m2 .
2. Now solve the exact same proglem using variables I, M and m instead of values.
Still keep the system frictionless. Look at your final answer and verify that it makes
sense.
3. Solve the problem again, this time with friction. Return to using the numbers given
above and take µ = 0.1.
4. Solve the problem with non-zero friction. Don’t use values but use the variables
I, M, m and µ.
5. Now put car A on an incline, with angle θ, and solve the problem using variables.
Look at your final answer and verify that it makes sense.
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