Download Lab Manual 2005

Data Analysis
Situation:
This is an exercise so that you may practice the skills of data analysis and interpretation.
You may need to refer to Appendix A, B and I. Appendix B contains technical information
on using Graphical Analysis.
Materials:
Graphical Analysis.
Procedure: Find a computer with Graphical Analysis either in one of the computer labs or the
physics lab. Everyone in the group must become an expert in this software, therefore
make sure everyone gets a turn at the computer.
Plot each data set and answer the associated questions. Remember to correctly label
each axis and find the appropriate line of best fit.
Data Set 1
Age
(months)
0
1
2
3
4
5
6
7
8
9
Data Set 2
Mass
(kg)
3.17
4.07
5.10
5.88
6.77
7.55
8.63
9.24
10.1
11.61
Diameter
(cm)
0.00
3.00
4.00
4.47
4.89
5.29
6.00
6.19
6.47
6.84
Area
(cm2)
0
7.07
12.57
15.85
18.75
21.99
28.31
30.09
32.90
36.75
Part 1 (Use Data Set 1)
1.
Identify the independent and dependent variables.
Independent: _______________________
Dependent: ________________________
2.
What is the name of the mathematical model that best describes this data?
(Refer to Appendix A)
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3.
Develop a specific physical equation that describes this data.
4.
Considering delta notation, write a sentence that describes what the slope represents.
5.
Evaluate the y-intercept with the 5% rule. Is this a zero intercept?
6.
Explain what the y-intercept represents.
7.
Develop a general physical equation that describes this data.
8.
Using both the graph and the specific physical equation, determine the mass at age 4.5
months. Was this found by interpolation or extrapolation?
9.
Using both the graph and the specific physical equation, determine the mass at age 360
months. Was this found by interpolation or extrapolation? [Manual scale the graph,
x: 0→ 400, y: 0→ 500 ]
10.
What do you think about the reliability of your answers in questions 8 and 9? Explain
your position.
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Part 2 (Use Data Set 2)
1.
Identify the independent and dependent variables.
Independent: _______________________
Dependent: ________________________
2.
What is the name of the mathematical model that best describes this data?
(Refer to Appendix A)
3.
Develop a specific physical equation that describes this data.
4.
Develop a general physical equation that describes this data.
5.
How can this data be linearized?
Linearize the data before answering the remaining questions in this section.
6.
Develop a specific physical equation for the linearized data.
7.
Evaluate the y-intercept with the 5% rule. Is this a zero intercept?
8.
Develop a general physical equation for the linearized data.
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9.
Compare the general physical equation found in question # 8 to that found in #4. How do
they compare? Which method is clearer or easier?
10.
A well known mathematical formula was used to create Data Set 2. Therefore it can be
proven that the slope of the linearized graph represents a known mathematical constant.
Determine the mathematical constant that the slope represents by examining the specific
physical equation of the linearized graph (#6) and comparing it with other known
equations. (Consider geometry and the variables which are being examined.)
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Kinematics(ULV)
Situation:
Consider driving on a long and straight section of highway using cruise control.
We could describe this as motion with a steady pace. (The speedometer always reads
some steady value.)
Now consider starting a car from a stoplight. How does this motion differ from
cruising along the Trans-Canada? Is stopping a car at a red light fundamentally different
from starting? Here we are dealing with an object whose speed is steadily changing.
The general purpose of this lab is to formally describe the different types of
motion. We are concerned with a description of “how” the object moves and are not
concerned with “why” it moves.
Materials:
A motion detector, metre stick, tractor, track, MPLI and Graphical Analysis.
Model 1 (Uniform Linear Velocity)
Purpose:
To find a mathematical model that describes the relationship between position and
time, for all objects with steady (pace) motion.
Procedure:
• Enter MPLI and open a file called “Kinematics.exp”. (For more information on using
MPLI see Appendix C.) Plug a motion detector into channel A on the MPLI box and
calibrate the motion detector following the instructions in Appendix C (Calibration may
have been completed by demonstrator.)
• To collect your data, place the tractor in front of the detector. Clear the surrounding
area of possible reflectors. Turn on the tractor and hold it in place at 40 cm from the
motion detector (start line).
• Have your partner click start on the computer and release the tractor when you hear
the motion detector collecting data. It is not necessary to stop the data collection, it stops
automatically at the end of the specified time interval.
• When the data collection is complete, draw a line of best fit for the graph.
→ click on Analyse in the pull down menu. Select Automatic Curve Fitting
and choose the model that best represents your graph (see Appendix A).
Remember, if the trend is horizontal use statistics.
Have your graph checked by an instructor. When you are given the OK, print the graph.
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Analysis:
1.
Record the slope and y-intercept of the position -time graph.
m=
b=
2.
Using delta notation, determine the general meaning of the slope of the position-time
graph.
3.
Determine the general meaning of the y-intercept of the position-time graph.
4.
Develop the specific and general equations for the position-time graph.
• Using the tangent and examine buttons, find the slope of the line of best fit at five arbitrary
time instants. This slope is called the instantaneous velocity. Record this data in the table
provided.
Arbitrary Instants
1st
2nd
3rd
4th
5th
Instantaneous Velocity
Time
• Once all the data has been recorded, minimize MPLI and open Graphical Analysis. Plot the
data collected and find the line of best fit.
5.
The slope of the velocity-time graph should be zero or very close to this. What does this
imply about the motion?
6.
Determine the general meaning of the y-intercept of the velocity-time graph.
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7.
Calculate the area under the velocity- time graph, from t2 to t5. (Note: The notation tn
represents the value of t for the nth arbitrary instant.)
8.
What does this area represent?
9.
How can you verify this interpretation using the position-time graph?
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Kinematics(ULA)
Situation:
Consider driving on a long and straight section of highway using cruise control.
We could describe this as motion with a steady pace. (The speedometer always reads
some steady value.)
Now consider starting a car from a stoplight. How does this motion differ from
cruising along the Trans-Canada? Is stopping a car at a red light fundamentally different
from starting? Here we are dealing with an object whose speed is steadily changing.
The general purpose of this lab is to formally describe the different types of
motion. We are concerned with a description of “how” the object moves and are not
concerned with “why” it moves.
Materials:
A motion detector, metre stick, dynamic cart, track, MPLI and Graphical
Analysis.
Model 2 (Uniform Linear Acceleration)
Purpose:
To find a mathematical model that describes the relationship between distance and
time, for all objects with steadily increasing motion.
Procedure:
• Enter MPLI and open a file called “Kinematics.exp”. Calibration may have been
completed by demonstrator.
• The simplest way to get an object to increase (or decrease) its speed at a “steady” rate
is to allow it to roll freely down (or up) an incline. We will use the collision cart rolling on
our track as the object and use the motion detector with the MPLI to collect our data.
• Set up the track with a slight incline. Place the detector at the high end and point it at
the cart. Clear the surrounding area of possible reflectors. To collect data, place your cart
40 cm in front of the detector (start line).
• Click on the start button and allow your object to roll down the track. Try a few
practice runs before collecting the data. The trick is to release the cart as you hear the
motion detector begin to click.
• When the data collection is complete, draw a line of best fit for the graph.
→ click on Analyse in the pull down menu. Select Automatic Curve Fitting
and choose the model that best represents your graph (see Appendix A).
Remember, if the trend is horizontal use statistics.
Have your graph checked by an instructor. When you are given the OK, print the graph.
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Analysis:
1.
Develop the specific equation for the position-time graph.
• Using the tangent and examine buttons, find the slope of the line of best fit at five arbitrary
time instants. This slope is the instantaneous velocity. Record this data in the table provided.
Arbitrary Instants
1st
2nd
3rd
4th
5th
Instantaneous Velocity
Time
• Once all the data has been recorded, minimize MPLI and enter Graphical Analysis. Plot the
data collected and find the line of best fit.
2.
Record the slope and y-intercept of the velocity-time graph.
m=
b=
3.
Using delta notation, determine the general meaning of the slope of the velocity-time
graph.
4.
Determine the general meaning of the y-intercept of the velocity-time graph.
5.
Develop the general and specific equations for the velocity-time graph.
6.
Calculate the area under the velocity- time graph, from t2 to t5. (Note: The notation tn
represents the value of t for the nth arbitrary instant.)
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7.
What does the area represent?
8.
How can you verify this interpretation using the position-time graph?
9.
Write the general physical equation for the position-time graph. (Compare specific values
of coefficients in question #1 and #5 to determine physical meanings of the A, B and C
coefficients.)
10.
Based on the velocity-time graphs obtained for both models, sketch the resulting
acceleration-time graph.
Model 1 (ULV)
Model 2 (ULA)
v
v
t
t
a
a
t
t
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Kinematics
Conclusion:
Make a concise summary of the properties of the graphs from the motion studied in both
models. Include tiny sketches, general equations, meanings of slopes, intercepts and areas.
type of motion
properties of x-t graph
properties of v-t graph
Constant velocity
- travelling with
“steady” (unchanging)
pace
Slope =
Yint =
General Eqn =
Slope =
Yint =
Area =
General Eqn =
Slope =
Yint =
General Eqn =
Slope =
Yint =
Area =
General Eqn =
Constant
acceleration
- travelling with
“steadily” changing pace
From now on we will refer to steady paced motion as Uniform Linear Velocity (ULV) and
steadily changing pace motion as Uniform Linear Acceleration (ULA).
Briefly sum up the results of this lab.
a. In your own words, state what it means to travel with constant acceleration and
constant velocity.
b. Explain how to calculate velocity based on a position -time graph.
c. Explain how to calculate acceleration based on a velocity-time graph.
d. Explain how to get displacement from a velocity-time graph.
e. Comment on any errors affecting your results.
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Projectile Simulation
Situation:
Projectile motion is simply a case of simultaneous motion in two directions. We
need to assess the motion in each direction.
Theory:
Divide and conquer. Analyzing what happens in two directions simultaneously is
difficult for us to do. However, if we can look at the vertical motion seperately from the
horizontal motion, then we have a chance at interpreting this event.
Procedure:
To aid in the study of projectile motion we will use a program called Working
Models 3.0.3 which will allow us to create and launch our own model. As well, the program
will allow us to analyse the resulting two-dimensional motion map. The following
instructions will help you create your own motion map.
→ Open Working Model 3.0.3 and maximize the window.
→ To create a reference frame for the motion of the object we need a co-ordinate system
→ enter View → Workspace → XY Axes
→ enter View → Workspace → Gridlines
→ To create an object click the circle located on the toolbar (about the third down), return
to the screen and “click and drag”. Try not to create an object that is to big because it
becomes more difficult to track.
→ Now we need to move our object to the origin and give it an initial velocity.
→ scroll down until the origin of the XY axes is in the lower left corner of the
window
→ double click on the object so that a “properties” window appears
→ set the x and y co-ordinates to 0 m. (Note that the object is now positioned at the
origin)
→ give the object an initial velocity by setting the horizontal and vertical
components. (Try vx = 5 m/s and vy = 7 m/s)
→ close the properties window
→ Now we need to track the motion of the projectile
→ enter World → Accuracy
→ in the box that appears change the animation step from the Automatic to 0.1s
→ enter World → Tracking → every frame
→ enter Measure → Time
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→ To measure the exact position of the projectile it is necessary to create a position box
→ click on the object
→ enter Measure → Position →All
→ in the Position box that appears click on “rot” to turn it off. This measurement is
not required since our object has no rotation.
→ We are now able to run the simulation and track the path taken.
→ click RUN (note: the RUN button will change to a STOP button once started)
→ after approximately 2.0s stop the motion and hit RESET
→ In the bottom left corner there appears a tracking device
Brings the objects back to the starting position (rewind)
Run the objects through the motion again
Advances the motion frame-by-frame (slow motion)
Rewinds the motion frame-by-frame
→ Advance the motion frame-by-frame by clicking
→ Collect the data required to analyse the motion. The x-axis represents the horizontal
motion and the y-axis represents the vertical motion.
→ keep in mind that the time for each point occurred at 0.100s intervals
→ record the x and y position of each point by advancing frame-by-frame and
reading the co-ordinates from the position box.
→ record all values in the data table provided.
→ Once all data has been collected minimize Working Models and enter Graphical
Analysis.
→ open a file marked “projecti.dat” (this contains a template of the necessary
graphs)
→ plot the position-time and velocity-time graphs for the horizontal and vertical
aspects of motion. You will obtain four graphs in all, two for horizontal
motion and two for vertical motion.
→ Remember to use the curve fitting options to obtain a line of best fit for each
graph.
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DATA
Dot #
Time (s)
Horizontal
x (m)
Vertical
y (m)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Analysis:
1.
Determine the specific physical equation for each graph. Use appropriate symbols for
each equation.
Position x vs. time :
Velocity x vs. time :
Position y vs. time :
Velocity y vs. time :
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2.
What motion model (ULV or ULA) applies to the horizontal motion? How do you know?
3.
What motion model applies to the vertical motion? How do you know?
4.
Referring to the appropriate graphs:
a) Determine the initial horizontal speed. How did you get this value?
b) Determine the initial vertical speed. How did you get this value?
5.
Using the values from question 4, calculate the launch velocity. Include a tip-to-tail
diagram and express the final answer in the form magnitude (units) @ degrees to the
horizontal.
6.
Use vector addition to determine the velocity at 0.300s.
7.
What should vy be at maximum height?
8.
Determine the velocity at the maximum height.
9.
What is the velocity at 1.5s ?
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10.
What is the velocity at 2.0s ? Show your workings. [Use the appropriate equations
determined in question 1.]
11.
What is the ball’s vertical and horizontal position at 2.0s ? Show your workings. [Use the
appropriate equations determined in question 1.]
12.
A golfer on Gros Morne drives golf balls off the peak at 14.0 m/s @ 20° above the
horizontal. If the peak is 800m above the bottom of the Fjord on the north side of the
mountain, then determine the ball’s:
a) time of flight
b) horizontal range
c) maximum altitude of the ball (relative to the Fjord)
d) impact velocity
13.
A ping pong ball bounces off a table, 0.85m above the ground. If the ball strikes the floor
at 5.44 m/s @ 56.5° below the horizontal, then determine the ball’s:
a) initial velocity
b) time of flight
c) horizontal range
d) maximum height relative to the floor
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- 18 -
Relative Velocity Simulation
Situation: Consider two ships at sea that are travelling near each other. The radar operator on
one ship can see the other ship on his radar screen. With each sweep of the radar it plots
the position of the other ship relative to his ship. His ship is located at the center of the
screen. From this information he can easily tell the rate at which the ship is approaching
him. This is known as relative velocity.
But does this tell us the other ships actual speed? Does it tell us the actual
direction the other ship is headed? This lab exercise will attempt to give you an
understanding of the concept of relative velocity and how it relates to actual velocity.
Frames of Reference
1.
Suppose you are on a ship that is moving “forward”. When you look over the side at the
water, which way does it appear to move? (Consider the speed of the water flow and the
direction in comparison to the speed of the ship itself)
2.
What if you were restricted to only looking at the water flowing along the side of the
ship? Do you think this observation could accurately give you information about the
motion of the ship?
3.
Suppose you were on a ship that was anchored in the middle of a river, where the current
is very fast. You look over the side and observe the water flow around the ship. What
does it look like?
4.
Re-examine your answer to question 2. Make any additional comments here.
5.
Have you ever had your car stopped at a traffic light and noticed the car ahead of you roll
back toward you. Did you briefly get the sensation that you are actually moving instead
of the car ahead? While this sensation is happening is there any way of determining for
sure who is actually moving?
6.
Now suppose that in question 5 you are not stopped at a light but are parked on a ferry
and as the ferry begins to pull out from the dock the car ahead of you, which was
mistakenly left in neutral, starts approaching you. Compare this situation with question
5. Which car is now moving? Is it possible to tell from your observations?
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Part 1:
One Dimensional Relative Velocity
Situation: To study one-dimensional relative velocity we will use Working Models to
create a motion map representing two ships travelling in the same direction along the
same linear path.
→ Open Working Models and maximize the screen.
→ In order to view the entire motion of our simulation we can now set the width of our
window.
→enter View → View Size → Window Width → 25 m
→ We will first need to create the objects used in the simulation. To make the
simulation easier to follow we will create two different types of objects (one
rectangular and one circular) of approximately the same size.
→ from the toolbar along the left side of the window click on the circle.
→ return to the window and "click and drag" to create a circle.
→ from the toolbar along the left side of the window click on the rectangle.
→ return to the window and "click and drag" to create a rectangle.
→ We will now have to set the initial properties of each object. Keep in mind that the two
objects will be travelling along the same linear path.
→ Double click on the rectangular object .
→ In the properties window that appears set the position and the velocity by changing to
the following co-ordinates x = -4.0 m, y = -2.0 m, vx = 3.0 m/s, and vy = 0 m/s.
→ In the pull down menu of the Properties window select the circular body (note that
the circular object now becomes highlighted).
→ We can now set the position and velocity of the circular object by changing to the
following co-ordinates x = -1.0 m, y = 1.0 m, vx = 2.0 m/s, and vy = 0 m/s.
→ Close the Properties window and note that the circular object should now be above
and slightly to the right of the rectangular object.
→ For our simulation we would like to create a “Bird’s Eye View”. Essentially, this
means that we would like to observe the motion of the objects as seen from above.
To accomplish this we must turn the gravity off.
→ enter World → Gravity → None
→ It is also useful to have the XY axes as a reference frame for the motion
→ enter View → Workspace → XY Axes
→ Note that since our view of the simulation occurs from above, the XY axes
now acts as a compass where the Y axes represents the North-South axes and
the X-axes represents the East-West axes.
→ Throughout our simulation it will be necessary to measure the exact position of each
object.
→ Click on the rectangular object.
→ Enter Measure → Position → All
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→ It is also convenient to have velocity vectors which indicate the magnitude and
direction of the velocity.
→ Enter Define → Vectors → Velocity
→ Click on the circular object and repeat the previous steps.
→ In the Position box that appears click on “rot” to turn it off. This measurement is not
required since our object has no rotation.
→ Move Position boxes to a more convenient location by clicking on the title and
dragging the box.
→ It will also be necessary to record the time of the motion as we run our simulation.
→ Enter World → Accuracy
→ Change the Animation Step to 0.100 s and click OK.
→ Enter Measure → Time
→ We are now able to measure the exact position of the two objects throughout
the run.
→ To run the simulation click the RUN button located in the upper left corner (note: The
RUN button becomes a STOP button once activated.)
→ After approximately 6 s stop the simulation (click STOP).
→ We are now able to collect data regarding the separation displacement between the two
objects.
→ In the bottom left corner there appears a tracking device
Brings the objects back to the starting position (rewind)
Run the objects through the motion again
Advances the motion frame-by-frame (slow motion)
Rewinds the motion frame-by-frame
→ Click RESET to return to the start of the motion.
→ Advance the motion frame-by-frame by clicking
→ For every 5 frames record the horizontal separation displacement (xc – xr) and the
time. [Note: We are using the “rectangle” ship as our reference object. That is, all of
our measurements are taken from the rectangle’s position.]
Horizontal Separation Displacement
(xc – xr)
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Time (s)
Analysis:
1.
Suppose you were on the bridge of the ”rectangle” ship. According to your observation,
how and in which direction does the other ship appear to be moving? (Consider how the
separation displacement is changing with time.)
2.
Suppose you were on the “circle” ship instead. How would the motion of the “rectangle”
ship appear? (Consider both magnitude and direction.)
→ To check your answers for #1 and #2 return to the simulation. It is possible to
run the simulation while using one of the objects as a frame of reference. That
is, the simulation will now be seen from the point-of-view of the chosen
object.
→ click on the "rectangle" ship
→ enter View → New Reference Frame → (the object name should
appear and the "show eyeball" should be turned on) → click Ok
→ run the simulation and note that the rectangle now appears fixed
while the circle and XY axes are in motion.
→ repeat for the "circle" ship
→ to return to the original reference frame enter View → Home
→ to reset the window width
enter View → View Size → Window Width → 25 m
3.
What is the separation displacement between the ships at the following time intervals?
t=0s
t = 1.0 s
t = 3.0 s
t = 5.0 s
t = 7.0 s
→ Minimize Working Models and enter Graphical Analysis. Plot a graph of separation
displacement vs. time.
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4.
Considering delta notation, determine the general meaning of the slope of the position time graph.
5.
What is the numerical value for the rate of change of separation displacement?
6.
What can you say about how the separation displacement is changing?
7.
Considering the initial velocities of each object, could the rate of change of separation
displacement be determined indirectly (without measurement from motion maps)?
Vc = _________
Vr = _________
8.
Now suppose that the same two ships are again travelling on the same path, but in
opposite directions, with the same speeds as they had before. Imagine you are on the
“rectangle” ship but this time the “circle” ship is ahead of you and heading towards
you. How would you determine the relative velocity of the circular ship?
Part 2:
Two Dimensional Relative Velocity
Situation: We will now use Working Models to examine the situation where two
moving objects, such as our two ships, are travelling in a straight line but not
along the same path.
→ Open Working Models by clicking on the minimized Working Models window (located next
to the start button).
→ Since we are entering back into the previous simulation the environment set-up will
be the same and we will only need to change the property settings of each object.
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→ We will now have to set the initial properties of each object. Keep in mind that the two
objects will not be travelling along the same linear path.
→ Double click on the rectangular object.
→ In the properties window that appears set the position and the velocity by changing to
the following co-ordinates x = -1.0 m, y = -5.0 m, vx = 2.0 m/s, and vy = 1.5 m/s.
→ In the pull down menu of the Properties window select the circular body (note that the
circular object now becomes highlighted).
→ We can now set the position and velocity of the circular object by changing to the
following co-ordinates x = -5.0 m, y = 2.0 m, vx = 2.5 m/s, and vy = -1.0 m/s.
→ Close the Properties window and note that the circular object should now be above
and slightly to the left of the rectangular object.
→ To run the simulation click the RUN button located in the upper left corner (note: The
RUN button becomes a STOP button once activated.)
→ After approximately 6 s stop the simulation (click STOP).
→ Objects should not collide. If a collision occurs, check co-ordinates and object size.
→ We are now able to collect data regarding the separation displacement between the two
objects.
→ Click RESET to return to the start of the motion.
→ Advance the motion frame-by-frame by clicking
→ For every 5 frames record the horizontal separation displacement (xc – xr), vertical
separation (yc – yr), and the time. [Note: We are using the “rectangle” ship as our
reference object. That is, all of our measurements are taken from the rectangle’s
position.]
Time (s)
Horizontal Separation Displacement
(xc – xr )
Vertical Separation Displacement
(yc – yr)
→ Once all data has been collected minimize Working Models and enter Graphical
Analysis.
→ In Graphical Analysis plot a displacement vs. time graph for both the horizontal
and vertical components. Do not forget to use the curve fitting options.
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Analysis:
1.
What does the slope of each graph represent?
2.
Calculate the velocity of the “circle” ship with respect to the “rectangle” ship in this 2-D
situation as you did in the 1-D situation. [Hint: Consider vector analysis.]
3.
Suppose you were on the bridge of the ”rectangle” ship. According to your observation,
how and in which direction does the other ship appear to be moving? (Consider how the
separation displacement is changing with time.)
4.
Suppose you were on the “circle” ship instead. How would the motion of the “rectangle”
ship appear? (Consider both magnitude and direction.)
→ To check your answers for #3 and #4 return to the simulation. It is possible to
run the simulation while using one of the objects as a frame of reference. That
is, the simulation will now be seen from the point-of-view of the chosen
object.
→ click on the "rectangle" ship
→ enter View → New Reference Frame → (the object name should
appear and the "show eyeball" should be turned on) → click Ok
→ run the simulation and note that the rectangle now appears fixed
while the circle and XY axes are in motion.
→ repeat for the "circle" ship
→ to return to the original reference frame enter View → Home
→ to reset the window width
enter View → View Size → Window Width → 25 m
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5.
Using the initial velocities assigned to each object, the velocity of each ship can be
determined to be:
Velocity of “rectangle” ship
Velocity of “circle” ship
2.5 m/s
2.7 m/s
@
@
36.9° N of E
21.8° S of E
.
.
.
Magnitude
2.5
2.7
θ
36.9º N of E
21.8º S of E
θ standard
Vector Difference
x-component
Vxr =
Vxc =
Vxc - Vxr =
y-component
Vyr =
Vyc =
Vyc - Vyr =
Find the vector difference between the velocities of the two ships.
6.
Compare the answers found in questions #2 and #5. What conclusion can now be made
regarding relative velocity?
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Constant Net Force Model
Situation:
At the heart of dynamics is the notion of net force. The constant net force model
says that when the net force on an object is non zero (Fnet ≠ 0), then the object
undergoes constant acceleration.
EX 1: Consider two identical cars stopped on a level road at a stoplight. The driver of car A is
very aggressive and will press the gas peddle to the floor when the light turns green. Car
B’s driver is more cautious and will only press the gas peddle a quarter of the way down.
Both drivers press on the gas as soon as the light changes.
What type of motion will each car have after the light changes colour?
What dynamic model applies to each car?
How might the motion of each car differ? Explain how and guess why this is so.
What variables might be displayed here? Which could be independent and which should be
dependent? What factor seems to be held constant?
EX 2: Now consider this. Two identical tractor trailers are stopped at the same intersection as
above. Both drivers are equally aggressive and will press on the gas all the way when the
light turns green. However, one truck is fully loaded with 2 tonnes of steel pipes and the
other is completely empty.
What type of motion will each truck have after the light changes colour?
What dynamic model applies to each truck?
How might the motion of each truck differ? Explain how and guess why this is so.
What variables might be displayed here? Which could be independent and which should be
dependent? What factor seems to be held constant?
- 27 -
Materials:
A motion detector, a force sensor (mounted on a dynamic cart), dynamic cart and
track, string, mass set, MPLI and Graphical Analysis.
Theory:
Before we continue with the experiment draw a semi scaled FBD and VCD, of a
dynamic cart with uniform linear acceleration on a level track.
cart
frictionless
m
What assumptions have you made about the forces acting on the cart?
Purpose:
To find a mathematical model that explains the relationship between net force,
mass and acceleration, for all objects with uniform linear acceleration.
Procedure:
• It should be clear that the rate at which you accelerate may be effected by more
than one variable. The apparatus we recommend you use, should work for all the
necessary tests.
• Enter MPLI and open the file called “Newton.exp”. A F vs. t and a v vs. t graph
should appear which will allow us to collect our data.
• Next you must calibrate the force sensor and motion detector (see Appendix C).
(Calibration may have been completed by demonstrator.) Make sure that when calibrating
the force sensor that the cart does not move and that the sensor is arranged in the way it
will be used.
• Since we have two independent variables we must run two tests to see how the
variables are related to the dependent variable. For each test, one variable must be held
constant while the other variables are being examined.
- 28 -
Part 1: Vary the force applied while keeping the mass of the cart constant.
→ Record the mass of the cart (include the mass of the attached force sensor).
m = ______________ kg
→ To vary the force applied, change the mass attached to the string. Vary the mass from
20g to 300g. For each trial record the mass attached and collect data.
→ Place the cart 40 cm from the motion detector (at start line). Attach a mass to the end
of the string and place string over frictionless pully.
→ Have your partner click start on the computer and release the cart when you hear the
motion detector collecting data.
→ Collect the required data and repeat procedure for new attached mass (force).
→ record the force applied from the force-time graph [Remember if the trend is
horizontal to use statistics and obtain the average value.]
→ record the acceleration from the velocity-time graph
→ Record all data in the table provided.
Part 2: Vary the mass of the cart while keeping the force applied constant.
→ Calculate and record the applied force using the equation given. ( mC is the mass of
the cart and mH is the mass attached to the string.)
FT =
mC mH g
mC + mH
FT = ______________ N
→ Vary the mass of the cart, in 100g increments, by placing masses on top of the cart.
Include the initial mass of the cart in all measurements.
→ Place the cart 40 cm from the motion detector (at start line). Attach a 50g mass to the
end of the string and place string over frictionless pully. Place mass on top of cart.
→ Have your partner click start on the computer and release the cart when you hear the
motion detector collecting data.
→ Collect the required data and repeat procedure for new cart mass.
→ record the acceleration from the velocity-time graph
→ Record all data in the table provided.
For each experiment do 7-10 trials. Record the data in table provided.
- 29 -
Part 1
Attached
Mass (kg)
Force (N)
Acceleration
(m/s2)
Part 2
Mass of Cart
(kg)
Acceleration
(m/s2)
When you have collected the data for all experiments, minimize the MPLI program and enter
Graphical Analysis.
→ Plot the data for each experiment. An acceleration vs. Force graph (for Part 1) and
acceleration vs. mass of cart graph (for Part 2) should be obtained.
→ Remember to use the curve fitting options to obtain a line of best fit for each graph.
Analysis:
1.
State the proportionalities for both graphs and the joint variation proportionality.
2.
Determine the average proportionality constant for the joint variation (see Appendix J).
What are its units?
3.
Rewrite the equation for this data.
a. Solving for acceleration.
b. Solving for force.
- 30 -
4.
According to 3b what are the units of force? Are these units equivalent to the newton?
5.
Based on the units given in question 4, describe in your own words what 1N of net force
is.
6.
Imagine a massless object under the influence of some net force. How much acceleration
does it have? [By massless we are referring to an object that is so small that it can be
considered massless but does not have a mass of zero.]
7.
A student told me that an object’s mass is its resistance to being in motion, ( i.e.,
massive objects come to a stop faster than lighter objects). Is this idea correct? Defend
your choice in terms of the graphs you have plotted.
8.
State two ways to calculate net force.
9.
Given this scale VCD and that the object has a mass of 2.50 kg, what is its acceleration?
How far will the object travel in 5 seconds, if it starts from rest?
FN
Fm
1 cm = 1 N
Fg sin θ
- 31 -
Fr
Fg cos θ
10.
An object accelerates at 9.80 m/s2 with a net force of 123.0 N, what is its mass?
11.
Given a net force of 15 N on a 1.25 kg cart, what is the acceleration?
12.
A 60 kg person sits on a level, carpeted floor. If the person is pushed across the floor with
a force of 255 N and the friction is known to be 245 N, determine the acceleration of the
person.
Conclusions:
Briefly sum up the results of this lab.
a. Compare the two ways of calculating net force.
b. State Newton’s second law in your own words.
c. State how are the variables related to each other.
d. Discuss any possible sources of error in this experiment.
- 32 -
Friction
Situation:
In the lab we create dynamic models in near perfect conditions. Namely we have
been avoiding this thing called friction. While we tested and proved that these models
work when friction is a factor, we did not look into the nature of this resistive force.
Since friction is a part of our everyday world, it deserves special attention. Physicists
refer to friction as a resistive force, what do you suppose this means?
Is friction always a bad thing? Explain.
EX 1: Consider a cardboard box (say for a new VCR) on a carpeted floor. You decide that you
will pull this box closer to the TV. Once the box is near the TV you can take the VCR out
of the box and put it on the TV stand. However, the empty box is in the way. This time
you will push the box away.
In both cases would you need a constant push or pull to keep the box moving? Explain why.
How did the pull on the full box differ from the push when it was empty? What factors may have
changed?
What about repeating this activity on a polished hardwood floor? Would you need to push as
hard? Explain your answer.
EX 2: Now suppose that you were moving a massive cardboard box with some ULV, by pulling
it across a carpeted floor. It is easier to move the box by pulling up at some angle to the
horizon than to apply a horizontal force. If the upward component of this pull is not large
enough to lift the box off the ground, sketch the vector component diagram.
Does the upward angled pull change the weight of the box?
Does the upward angled force change any other force acting on the box?
What other factors might affect friction?
- 33 -
Materials:
A force sensor, wooden blocks, surface board, mass set, MPLI and Graphical
Analysis.
Purpose:
To find a mathematical model that explains the relationship between weight,
normal force, and friction, for all sliding objects.
Procedure: We want you to examine the nature of sliding friction. To do this you will pull the
wooden block along some surface with a calibrated force sensor. Each group will use a
different surface combination. For all tests that you might do you must first devise a way
to knowingly measure the sliding friction force. Remember that the force sensor only
measures the force you supply to pull the object.
When pulling the block, what type of motion should you strive for? Justify in terms of your
dynamic models? Draw a semi scale FBD and VCD.
• Enter MPLI and open the file called “Friction.exp”. A F vs. t graph should now appear. Next
you must calibrate the force sensor (see Appendix C). Make sure that when calibrating the force
sensor, it is arranged in the way it will be used.
• In this experiment we will vary the normal force (by adding mass to the block) and record
both the kinetic and static friction.
• Where possible do several trials using different values of the normal force. Remember to pull
the block with constant velocity, starting from rest with no tension in the string.
• Record all data in the table.
→ calculate the normal force
→ record the value of friction when the object was moving (kinetic) and the maximum
value of friction before the object started to move (static) from the Force- time graph.
Combined mass of
block and attached
mass (kg)
Normal Force (N)
- 34 -
Static Friction (N)
Kinetic Friction (N)
• When you have collected the data, minimize the MPLI program and enter Graphical Analysis.
Plot the data for each experiment. On the same graph plot maximum efforts to start moving (fs)
and average effort to keep moving (fk) vs. the normal force.
Analysis:
1.
Based on the pre-lab questions, which factor is best linked with friction, Normal Force or
Weight?
2.
State the nature of the relationship between friction and the normal force.
3.
Develop the general and specific equation of both the static vs normal force and kinetic
vs. normal force graphs.
4.
What is the value and general meaning of the slope of your friction vs. normal force
graph. (Consider unit analysis.)
5.
When friction is involved, does it take more force to start motion then it does to keep
moving? What does this say about the nature of friction?
6.
Air resistance is another form of friction. Move your arm through the air with a low
uniform velocity. Now imagine if your arm is stuck out of the window of a car moving at
100 km/h. When does air resistance feel stronger? What does air resistance depend on?
Would our model of sliding friction apply to explain air friction? Explain.
- 35 -
7.
Given that two objects of equal mass and surface texture but different contact areas, are
sliding across the same surface with constant velocity, which is true:
a.
The block with greater contact area feels a larger frictional force.
b.
The block with smaller contact are feels a greater frictional force
c.
Both block feel equal fictional force
(Justify your choice. Try examining your block using different surface areas.)
8.
A 3.2 kg block is allowed to slide down a rough but perfectly vertical surface. What is the
frictional force on the block? Explain?
9.
How much force is required to accelerate a 30 kg block, at 2.5 m/s2, along your surface,
when it is horizontal? Include a semi scale FBD and VCD.
10.
What is the frictional force experienced by a 15.0 kg block sliding on your horizontal
surface?
Conclusions:
Briefly sum up the results of this lab.
a.
What factors govern the nature of sliding friction?
b.
What factors determine the magnitude of friction? What ones do not?
c.
State differences in static and kinetic friction.
d.
Discuss any sources of error which may have occurred.
- 36 -
Circular Motion
Situation:
Consider an object like a rubber ball tied to a piece of string being swung in a
horizontal circular path at a constant speed.
Is this type of motion natural? That is, if the string broke would the ball continue in its
circular path?
What is the string doing to the ball that causes it to move the way it does?
Predict how changing the ball’s mass will affect the tension in the string?
Predict how changing its speed will affect the tension?
Predict how changing the radius of the path will affect the tension?
Theory:
Objects moving in a circular path require a force, known as the centripetal force,
to keep the object moving along that path. Centripetal force is a center seeking force. The
magnitude of the centripetal force is affected by the object’s speed, mass, and the radius
of the circle in which the object is travelling.
Purpose:
We would like to develop a mathematical model that shows the effect of the
objects speed, the radius of the path, and the mass of the object on the amount of force
required to cause it to move in a horizontal circular path.
Materials:
MPLI, turntable, Force sensor, various masses, and string.
Procedure: To develop this model we have designed a turntable that allows you to record the
force while varying the velocity, radius, and mass of the object. However, to successfully
determine the relationship we must examine the effect that each variable has on the force
while the other variables remain constant. Therefore, we have divided the lab into three
parts, where different groups will perform different parts and the information will be
shared to obtain the final result.
- 37 -
Part 1: Varying the mass.
•
Enter MPLI and open the file “circular.exp”. This file contains a Force vs. Time graph,
which will allow you to record the force, and also contains the necessary calibration.
•
While varying the mass it is important to keep the radius of the circular path constant.
The radius should be measured from the center of the turntable to the center of the object.
The radius can be adjusted by moving the force sensor up or down the ring stand. Record
the radius that you are using.
r=
•
It is also important to keep the velocity of the object constant. Start your turntable using
a gentle push. If the same person starts the motion each time with a steady and consistent
force then the velocity should remain fairly constant. However, there will be some
variation.
→ As the object rotates a sin wave appears on the Force vs. Time graph.
→ The time required for one complete rotation is the time difference between
successive peaks. Therefore, we can keep the velocity constant by allowing the
sweep to run until we have a certain number of waves for each trial (example: 5
waves in 5s).
→ Once the correct number of waves is reached press stop or the readings will
continue and the necessary information will be lost. It is important to stop at the
end of the sweep.
→ We can now calculate the velocity since we know the distance
travelled by the object in one cycle and the time required for that cycle to be
completed. (Note: time for one cycle = time / # of waves, the distance
travelled is equal to the circumference of the circle.)
average velocity =
•
Record the force from the graph by taking the statistical average. Both the force and the
mass should now be recorded in an appropriate table. Repeat the experiment several times
using different masses. Determine the mass of your object using the balance provided.
Force (N)
•
Mass (Kg)
Once all data has been collected enter Graphical Analysis and plot a graph of Force vs.
Mass. Linearize the data to determine how the force and mass are related. Print three
copies of your graph.
- 38 -
Part 2: Varying the Velocity.
•
Enter MPLI and open the file “circular.exp”. This file contains a Force vs. Time graph,
which will allow you to record the force, and also contains the necessary calibration.
•
While varying the velocity it is important to keep the radius of the circular path
constant. The radius should be measured from the center of the turntable to the center of
the object. The radius can be adjusted by moving the force sensor up or down the ring
stand. Record the radius which you are using.
r=
•
It is also important to keep the mass of the object constant. Record the mass of your
object using the triple beam balance provided.
m=
•
Start your turntable using a gentle push. By varying the force each time then a different
velocity should be obtained.
→ As the object rotates a sin wave appears on the Force vs. Time graph. The time
required for one complete rotation is the time difference between successive peaks.
→ We can now calculate the velocity since we know the distance travelled by
the object in one cycle and the time required for that cycle to be completed. (Note:
time for one cycle = time / # of waves, the distance travelled is equal to the
circumference of the circle.)
•
Record the force from the graph by taking the statistical average. Both the force and the
velocity should now be recorded in an appropriate table. Repeat the experiment several
times using different velocities.
Force (N)
•
Velocity (m/s)
Once all data has been collected enter Graphical Analysis and plot a graph of Force vs.
Velocity. Linearize the data to determine how the force and velocity are related. Print
three copies of your graph.
- 39 -
Part 3: Varying the radius.
•
Enter MPLI and open the file “circular.exp”. This file contains a Force vs. Time graph,
which will allow you to record the force, and also contains the necessary calibration.
•
While varying the radius it is important to note that the radius of the circular path should
be measured from the center of the turntable to the center of the object. The radius can be
adjusted by moving the force sensor up or down the ring stand. It is also important that
the radius start from the smallest and increase at equal increments to the largest
obtainable radius.
•
It is also important to keep the mass of the object constant. Record the mass of your
object using the triple beam balance provided.
m=
•
It is important to keep the velocity of the object constant. Start your turntable using a
gentle push. Since the radius of the object’s path is increasing than the time interval for
each reading will also have to increase for the velocity of the object to remain constant.
→ As the object rotates a sin wave appears on the Force vs. Time graph. The time
required for one complete rotation is the time difference between successive
peaks. Therefore, we can determine the time interval by recording the number of
waves in a certain amount of time (example: 12 waves in 8s).
→ For the initial reading, with smallest radius, try to get the table moving as
quickly as possible. Keep trying until 12 peaks appear in the 8s time interval. We
can now calculate the velocity since we know the distance travelled by the object
in one cycle and the time required for that cycle to be completed. (Note: time for
one cycle = time / # of waves, the distance travelled is equal to the
circumference of the circle.)
→ For each additional reading, the radius will increase. To ensure v is constant
for all trials then we must also increase the time period (T) by the same amount.
v = 2πr
T
← increase r by a factor of n
← increase T by same factor of n
* to increase T by a factor of n, we need
to change the # of cycles by factors of 1/n.
guideline:
Radius
# of cycles in 8s sweep
r1 = 5cm
r2 = 2( r1 ) = 10cm
r3 = 3( r1 ) = 15cm
r4 = 4( r1 ) = 20cm
r5 = 5( r1 ) = 25cm
r6 = 6( r1 ) = 30cm
→
→
→
→
→
→
- 40 -
12
½ (12) = 6
1
3 (12) = 4
¼ (12) = 3
1
5 (12) = 2.4
1
6 (12) = 2
→
•
For each additional reading, allow the sweep to run until we have the correct
number of waves for the 8s time frame. Once the correct number of waves is
reached record the average force.
Record the force from the graph by taking the statistical average. Both the force and the
radius should now be recorded in an appropriate table. Repeat the experiment several
times using a different radius each time.
Force (N)
•
Radius (m)
0.05
0.10
0.15
0.20
0.25
0.30
Once all data has been collected enter Graphical Analysis and plot a graph of Force vs.
Radius. Linearize the data to determine how the force and radius are related. Print three
copies of your graph.
Analysis:
Once you have obtained all three graphs (one for each variable) determine the
proportionalities for each. You can now determine the joint variance (see
Appendix J) and use your information to calculate the proportionality constant.
- 41 -
1.
A 25.0 kg child on a merry-go-round is moving with a speed of 1.35 m/s when 2.0 m from
the center of the merry-go-round. Calculate the net horizontal force exerted on the child.
2.
A horizontal force of 200 N is exerted on a 2.0 kg disc as it is rotated uniformly in a
horizontal circle of radius 1.0 m. calculate the speed of the disc.
3.
A 0.40 kg ball, attached to the end of a cord, is rotated in a circle of radius 1.2 m on a
frictionless horizontal surface. If the cord will break when the tension exceeds 50 N, what
is the maximum speed the ball can have?
- 42 -
Collisions
Situation:
Have you ever witnessed a car accident? How many different types of collision
have you seen? Most people have witnessed various types of collisions, either real life or
movie stunts, and can describe different situations. For example, imagine a speeding car
colliding with a parked car. After impact the two vehicles could bounce apart and move
off at different velocities or they could become entangled and move together at one
combined speed. Now consider a head on collision where the two vehicles, moving
towards each other, stick together after impact and continue with a new velocity.
Each of these collisions, although having different conditions before and after impact, can
be analyzed using the same techniques since collisions between objects are governed by
the laws of momentum and energy. In this lab we will examine various collisions using
both the Law of Conservation of Momentum and the Law of Conservation of Energy.
Theory:
The Law of Conservation of Energy allows us to analyze initial conditions for a
system and predict conditions and behaviour at a later time. The law states that the energy
in a system is constant. However, some energy may be in a form that isn’t easy to
measure – for instance, frictional heating or the deformation of an object. The Law of
Conservation of Momentum provides a simple method for the analysis of systems that
include these elusive forms of energy.
The Law of Conservation of Momentum states that the total momentum of a system is
constant. Momentum is the “quality” or “durability” of motion. In physics terms,
momentum is an object's mass multiplied by its velocity. The symbol for the quantity
momentum is the small case “p”; therefore the equation can be written as:
p=m× v
It is important to note that the law of conservation of momentum states that the total
momentum in a “system” is constant. When we use the word “system”, we mean all of
the objects we are considering. We could have several objects that interact with each
other, but with nothing else. In that case, one object could transfer some or all of its
momentum to another object, but the sum of all of the momentum in this system would
remain the same. However, a force acting for a given amount of time will change an
object's momentum.
Put another way, an unbalanced force always accelerates an object - either speeding it up
or slowing it down. If the force acts opposite the object's motion, it slows the object
down. If a force acts in the same direction as the object's motion, then the force speeds
the object up. Either way, a force will change the velocity of an object and if the velocity
of the object is changed, then the momentum of the object is changed.
- 43 -
These concepts are merely an outgrowth of Newton's 2nd law. Newton's 2nd law
( Fnet = m × a ) stated that the acceleration of an object is directly proportional to the net
force acting upon the object and inversely proportional to the mass of the object. When
v − v0
combined with the definition of acceleration ( a =
), the following equalities result.
t
FNet = m
∆v
t
If both sides of the above equation are multiplied by the quantity t, a new equation
results.
FNet t = m ∆v
FNet t = ∆p
To truly understand the equation, it is important to understand its meaning in words. In
words, it could be said that the force times the time equals the mass times the change in
velocity. In physics, the quantity FNet × t is known as the impulse. Since the quantity m×v
is the momentum, the quantity m×∆v must be the change in momentum. The equation
really says that the
Impulse = Change in momentum
In a collision, an object experiences a force for a specific amount of time which results in
a change in momentum (the object's mass either speeds up or slows down). The impulse
experienced by the object equals the change in momentum of the object. During the
collision the two objects are in contact for the same amount of time and exert equal, but
opposite, forces on each other (Newton’s 3rd Law). As a result the total momentum of the
system is conserved.
The system we wish to focus on consists of two dynamic carts that will be used to
recreate various types of collisions. In some collisions, objects bounce off of each other.
If no energy is used to deform the object (or produce sound or heat), these collisions are
termed elastic collisions. (In other words, if kinetic energy is conserved in a collision,
then it is elastic.) In other collisions, objects still bounce, but use some of their energy
deforming each other and/or producing sound. These are called inelastic collisions.
Sometimes objects stick together during a collision. These are called completely inelastic
collisions. Examples of completely inelastic collisions include throwing silly putty
against a concrete wall, colliding a car with a tree, or rolling a dynamics cart into another
dynamics cart and allowing the velcro on each to stick the carts together.
Again the momentum and energy of the system are conserved in all types of collisions.
However, the form of the final energy is not always easy to measure. Objects in inelastic
collisions lose kinetic energy to the energy of deformation, sound, or heat. For this
reason, we use the Law of Conservation of Momentum when examining collisions.
- 44 -
In the lab, we will measure the velocity of two carts on dynamic tracks using the LabPro
Software for various collisions. From this we can determine their respective momentum.
In principle, we could make systems that have more than two cars, but that would just
complicate the experiment and the calculations. Remember that if the cars bump into
anything, like the ends of the track, momentum is no longer conserved, since the car has
interacted with something outside the system.
Purpose:
The principles of conservation of momentum and/or energy are studied for both
elastic and inelastic purposes.
Materials:
LabPro software, dynamic carts and tracks, vernier motion sensors, balance.
Part 1
Procedure: The following experiment will be demonstrated by the instructor. Students must
pay close attention and record all data in the table provided.
Set up a collision between a dynamic cart and a force sensor. The force sensor will
determine the force upon impact. Use the statistics option to determine the average force
applied during contact. Also the duration of the contact (collision) can be read from the
time axis. There is also a column in the table to record the area beneath the F vs t graph
for the duration of the collision.
mass
vi
vf
∆p
Ave Force
∆t
Impulse
Cart
Analysis:
1.
How does the change in momentum for the cart compare to the impulse?
2.
What causes the change in momentum for the cart?
3.
What does the area under the F vs. t graph represent?
4.
How does the area recorded compare to the impulse?
- 45 -
Area
Part 2
Procedure: Collisions between freely moving carts are executed on the tracks. Friction
between the track and the carts are assumed negligible. The acceleration due to the force
of gravity is eliminated by restricting all motion to the horizontal plane ( i.e., levelling the
track). The track will be levelled in advance by the lab instructor.
Of six different possible collisions, two or three collisions will be assigned for
examining. For each collision set-up, record the mass of both carts using the balance
provided.
Determine the initial and final velocities for both carts from the graphs obtained and
record all information in the table provided. An explanation as to how to read from the
graphs will be provided by the lab demonstrator.
Collision:
Mass
(kg)
Vi (m/s)
Vf (m/s)
Pi
(kg m/s)
Pf
(kg m/s)
Cart 1
Cart 2
Total momentum of Cart 1 and Cart 2
(Σ)
Compare the initial and final total momentum of the two-cart system.
→ Has momentum been conserved? If not, why?
Collision:
Mass
(kg)
Vi (m/s)
Vf (m/s)
Pi
(kg m/s)
Pf
(kg m/s)
Cart 1
Cart 2
Total momentum of Cart 1 and Cart 2
(Σ)
Compare the initial and final total momentum of the two-cart system.
→ Has momentum been conserved? If not, why?
- 46 -
Collision:
Mass
(kg)
Vi (m/s)
Vf (m/s)
Pi
(kg m/s)
Pf
(kg m/s)
Cart 1
Cart 2
Total momentum of Cart 1 and Cart 2
(Σ)
Compare the initial and final total momentum of the two-cart system.
→ Has momentum been conserved? If not, why?
Possible Collisions
A.
A light cart (m) travels toward a heavy cart (M), which is at rest. The two collide
and do not stick together.
B.
A light cart (m) travels toward a heavy cart (M), which is at rest. The two collide
and stick together.
C.
A light cart (m) and heavy cart (M) travel toward each other. The two collide and
do not stick together.
D.
A light cart (m) and heavy cart (M) travel toward each other. The two collide and
stick together.
E.
A light cart (m) is moving toward a heavy cart (M), which is also moving in the
same direction (chase situation). After the collision both carts stick together.
F.
A light cart (m) and heavy cart (M) are both at rest and placed end-to-end. When
the release pin is pushed they shoot apart.
- 47 -
Part 3
Procedure: Referring to the collisions performed in part 2, complete the following table and
answer all questions.
Collision 1:
Mass
(kg)
Vi (m/s)
Vf (m/s)
KEi
(J)
KEf
(J)
Cart 1
Cart 2
Total Kinetic Energy of Cart 1 and Cart 2
(Σ)
Compare the initial and final total kinetic energy, (KEi to KEf), of the two-cart system.
→ Is the kinetic energy conserved? If kinetic energy is not conserved, where
does it go? (What form has it been converted to?)
→ Find the amount of KE lost in the collision.
→ Is the collision elastic or inelastic?
Collision 2:
Mass
(kg)
Vi (m/s)
Vf (m/s)
KEi
(J)
KEf
(J)
Cart 1
Cart 2
Total Kinetic Energy of Cart 1 and Cart 2
(Σ)
Compare the initial and final total kinetic energy, (KEi to KEf), of the two-cart system.
→ Is the kinetic energy conserved? If kinetic energy is not conserved, where
does it go? (What form has it been converted to?)
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→ Find the amount of KE lost in the collision.
→ Is the collision elastic or inelastic?
Collision 3:
Mass
(kg)
Vi (m/s)
Vf (m/s)
KEi
(J)
KEf
(J)
Cart 1
Cart 2
Total Kinetic Energy of Cart 1 and Cart 2
(Σ)
Compare the initial and final total kinetic energy, (KEi to KEf), of the two-cart system.
→ Is the kinetic energy conserved? If kinetic energy is not conserved, where
does it go? (What form has it been converted to?)
→ Find the amount of KE lost in the collision.
→ Is the collision elastic or inelastic?
Questions:
1.
Why can you ignore the potential energy of the carts?
- 49 -
2.
As a ball falls toward earth, its momentum increases. How would you reconcile this fact
with the law of conservation of momentum?
Conclusion:
Briefly sum up the results of this lab.
a.
Identify the different types of collisions.
b.
Which method is the preferred method for analyzing collisions?
c.
How does an applied force effect the momentum of an object?
d.
Were there any possible sources of error in this experiment?
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Energy
Situation:
You’ve no doubt often heard people say, “I’d like to do some work but I don’t
have the energy.” From this context we understand what they are trying to say but you
may find it difficult to define what is meant by the term “work” or “energy”. These terms
have a meaning in a scientific sense as well although they are more abstract.
You will learn in class that work is associated with a force acting on an object to
move it some distance and that energy, which is thought of as the ability to do work, can
be related to mass and velocity. In the following lab we will verify the relationship
between work and energy while examining the Work-Energy Theorem and Conservation
of Energy.
Materials:
Motion detector, force sensor, dynamic cart and track, string, mass set, MPLI and
Graphical Analysis.
Part 1: Work - Energy Theorem
Procedure:
→ Enter MPLI and open the file AEnergy.exp@. A Force vs. time, position vs. time, and
velocity vs. time graph should appear. All necessary calibrations have been
completed.
→ Record the mass of the cart (include the mass of the force sensor).
m=
→ Set up the apparatus as shown in figure 1.
Motion
sensor
Cart and
Force sensor
Xo
X1
X2
50g
mass
Figure 1
Ensure that the cart is placed 40 cm from the motion sensor (at start line) and a 50g mass
is placed on the end of the string.
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→ Run the experiment to collect data. [Remember to release the cart when the ticking of
the motion sensor can be heard.]
→ Record the tension in the string by taking the average from the force vs. time graph.
FT =
→ From the velocity vs. time graph, determine the velocity of the cart at three separate
positions. Record both the position (x) and velocity (v) of the cart in the table
provided.
position-x
(m)
velocity
(m/s)
0
1
2
→ Verify the Work-Energy Theorem for positions x1 and x2
Wnet = ∆KE
x1 : compare position 1 & 0
x2 : compare position 2 & 1
- 52 -
Part 2: Conservation of Energy
Procedure:
→ Continue to use the same MPLI program. Set up the apparatus as shown in figure 2.
Motion
Sensor
1
x0
x1
x2
h2
Cart
h1
h0
Figure2
Remember to place cart 40 cm from motion sensor (at start line). Note that the force
sensor and attached mass are not required.
→ Record mass of cart. ( Note: the force sensor is not required and can be removed.)
m=
→ Place the cart at height ho and run the experiment to collect data. [Remember to
release the cart when the ticking of the motion sensor can be heard.]
→ From the velocity vs. time graph, determine the velocity of the cart at three separate
positions. Record both the position (x) and velocity (v) of the cart in the table
provided.
position-x
(m)
velocity
(m/s)
height
(m)
0
1
2
→ Locate, on the track, each position recorded. Then, using a ruler, measure the
corresponding height of the track at these positions and record all values in the table
provided.
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→ Verify Conservation of Energy where
PEo + KEo = PE1 + KE1 = PE2 + KE2
→ Repeat Part 2 with 500g mass placed directly on the cart.
position-x
(m)
velocity
(m/s)
0
1
2
PEo + KEo = PE1 + KE1 = PE2 + KE2
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height
(m)
Part 3: Non-conservative System
Procedure:
→ Continue to use the same MPLI program. Set up the apparatus as shown in figure 3.
X2
X1
X0
Motion
Sensor
Cart and
Force Sensor
h2
100g
mass
h1
ho
Figure 3
Ensure that the cart is placed 40 cm from the motion sensor (at the start line) and a 100g
mass is placed on the end of the string.
→ Record the mass of the cart (include the mass of the force sensor).
m=
→ Run the experiment to collect data. [Remember to release the cart when the ticking
of the motion sensor can be heard.]
→ Record the tension in the string by taking the average from the force vs. time graph.
FT =
→ From the velocity vs. time graph, determine the velocity of the cart at three separate
positions. Record both the position (x) and velocity (v) of the cart in the table
provided.
position-x
(m)
velocity
(m/s)
0
1
2
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height
(m)
→ Locate, on the track, each position recorded. Then, using a ruler, measure the
corresponding height of the track at these positions and record all values in the table
provided.
→ Verify that the non-conservative work = ∆ET where
WNC = ∆KE + ∆PE
- 56 -
Torque
Situation:
Have you ever used a wrench to loosen a nut that was rusted or corroded? The
wrench is a tool that can generate a twisting action on the nut (which we will refer to as
the “pivot”). In physics we refer to a term called “Torque” which is kind of difficult to
define conceptually with good clarity. Probably the best way to define torque is to say
that it is an entity that attempts to twist or rotate an object. The use of the word “entity”
leaves a lot to the imagination but comparing torque with force may be helpful. Just as
forces attempt to change an objects linear velocity, torque attempts to change an objects
angular velocity (rate of rotation).
1.
What do you supply to make the wrench twist around the nut?
2.
Does the location of your hand on the wrench handle play an important role in
determining how strong a twisting action is generated?
3.
If the nut was stuck really tight what changes could we try to help generate a stronger
twisting action?
PART 1
Purpose:
To develop a mathematical model (equation) that expresses how Torque is related
to Force, and Lever arm length. (Angle of the force kept constant and at 90° to the
lever arm)
Materials:
A metre stick, lever holder, string, a force sensor, MPLI and Graphical Analysis
Procedure: For this test, take your metre stick as a lever and place a lever holder exactly at
the balance point and place it on the pivot. Have the right side of the lever extended over
the edge of the bench.
- 57 -
• We need to provide a “controlled” torque that we will attempt to balance with our force sensor’s
applied force. This “controlled” torque can be provided by hanging a fixed mass of 200 g, 20 cm
away from the pivot on the left side. We do not yet know the magnitude of the “controlled” torque
but we do know that it is being kept constant throughout all the trials.
• Attach a force sensor to the MPLI box, enter MPLI and open a file called “Torque.exp”.
Calibrate the force sensor (see Appendix C).
• Tie a string to the calibrated force sensor and, starting 5 cm from the pivot, apply a perpendicular
downward force until the meter stick is balanced. Remember we must keep the string
perpendicular to the lever arm.
• Click start and measure the balancing force required. Record the force required and the length of
the lever arm in the table provided.
• Repeat this process while increasing the distance from the pivot at 5 cm intervals. Record the
balancing force required and the length of the lever arm for each trial.
Force (N)
Lever arm length (m)
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
• Minimize MPLI and enter Graphical Analysis to plot a graph of F vs L. Linearize and record the
slope and y-intercept.
Slope =
y-intercept =
Analysis:
1.
Write the specific and general equations for the linear graph, solving for slope.
2.
Describe what the slope represents. (Consider units)
3.
Can we compare the value of the slope to any other constant quantities in the experimental
apparatus? (Consider how the “controlled” torque was established).
- 58 -
Part 2
Purpose:
To develop a mathematical model (equation) that expresses how Torque is related to
Force, Lever arm distance, and the angle at which the force acts.
Consider the diagram of the hand and wrench.
1.
Suppose the hand pushed directly towards the right. What could be said about the amount of
twisting action?
2.
Suppose the hand in the diagram pulled directly towards the left. What could be said
about the amount of twisting action?
3.
What could be said about the amount of twisting action, if the hand were at an angle
between 0 and 90°?
4.
In question 3, the force from the hand would have a vertical and horizontal component.
How would you calculate these components?
5.
Which component is responsible for the torque?
Procedure:
With the “control” torque kept at the same constant as in test 1, measure the force
required to balance the meter stick if the string was at 60° to a 35 cm lever arm.
Remember to keep the angle between the force sensor and the string at 90°.
F = _______________
θ = ______________
l = _____________
Analysis:
1.
Is changing the angle of our pulling force so that it is not perpendicular to the handle
influential on the amount of force applied?
2.
Determine the components of the applied force and then calculate the torque.
τ=
Fx =
Fy =
- 59 -
3.
Can you rewrite the equation from the previous section (question 3), to take the angle of
action into account?
Part 3
Purpose:
In the previous experiment we developed the notion of torque as the product of a
force perpendicular to a lever and the length of the lever arm. We did this by keeping a
lever balanced, i.e., no rotation around the pivot. That is the clockwise torque equalled the
counter clockwise torque. In part 3 we explore this aspect of equilibrium by setting up a
system with an unbalanced lever and then calculating the total torque acting on it.
Materials:
1 metre stick, a lever clamp without stirrup, 2 clamp holders with stirrups, triple
beam balance, hanging masses and a lever stand.
Procedure:
• Place the lever clamp without stirrup on the 60cm mark of the metre stick and then place
it in the lever stand.
→ Provide a controlled torque by placing a 200g mass at 20cm from the
pivot, on the short side.
→ Determine where, on the long side, a 100g mass would have to be placed to balance the
metre stick.
• Make a detailed drawing of your balanced lever.
• Calculate and tabulate all clockwise and counter-clockwise torque around the pivot.
(Note that a combined mass for the weight and stirrup must be used for all calculations.)
Are the clockwise and counterclockwise torques balanced? If not, why?
- 60 -
Analysis:
1.
Center of gravity is the notion that on 3-D objects gravity acts on a point deep inside. Your
instructor can show you how to find the center of gravity. Find and record the center of
gravity.
2.
Calculate the torque caused by the stick itself. Does this account for the discrepancy?
3.
Make a detailed drawing of your balanced lever including the correction for the discrepancy.
Conclusion:
Briefly sum up the results of this lab.
State how the angle at which force is applied influences the torque.
Identify the component of force which causes rotation.
State the conditions of rigid body equilibrium.
Were there any possible sources of error in this experiment?
- 61 -
- 62 -