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1206 - Concepts in
Physics
Monday, November 23rd 2009
Notes
• Midterms (round 2) are ready to be picked
up in my office
• Today : 1:00 pm - 2:30 pm or after 4:00 pm
• Tomorrow before 11:30 am
• Wednesday 8:00 - 10:00 am or after
11:30 am
Notes
• Wednesday - assignment #7 is due
• Wednesday - last 10 min for evaluations, I
need a volunteer to be the returning officer
Notes
•
•
Preparations for final exam (2.5 weeks from now)
•
I will add a few more things to the webpage
- a new formula sheet
- midterms (first and second round)
- summary of math needed
- list of exercises from the textbook
•
We will spent one more lecture next week (likely
Friday) doing some example calculations/problems
•
We finish the topics for final exam next
Wednesday (inclusive)
We will start today by going over some of the
second round of midterm questions
Notes
•
The last week before the final exam is a busy time
for me, since we have collaboration meeting(s) and
I will be at SNOLAB to participate in them
•
Therefore, the time you will be able to find me in
my office are limited. So here are my office hours/
absences for these days. (Email is possible)
•
Thursday, December 3rd :
Friday, December 4th
:
Monday, December 7th
:
Tuesday, December 8th
:
Wednesday, December 9th :
not available
8:00 am - 10:00 am
8:00 am - 10:00 am
not available
8:00 am - 10:00 am
Multiple choice
1.) Which of the following quantities is not a scalar?
a)Energy
b)Work
c)Force
d)Power
2.) One movable pulley does
a)change direction only
b)distribute the forces equally to each rope
c)nothing
d)leave only one third of the force on the rope you are pulling on
3.) Archimedesʼ principle describes the relation between
a)Pressure and area
b)Energy components for fluids
c)the magnitude of the buoyant force and the weight of the
displaced fluid
d)the displaced fluid and the probability for an object to swim
Multiple choice
1.) Which of the following quantities is not a scalar?
a)Energy
b)Work
c)Force
d)Power
2.) One movable pulley does
a)change direction only
b)distribute the forces equally to each rope
c)nothing
d)leave only one third of the force on the rope you are pulling on
3.) Archimedesʼ principle describes the relation between
a)Pressure and area
b)Energy components for fluids
c)the magnitude of the buoyant force and the weight of the
displaced fluid
d)the displaced fluid and the probability for an object to swim
Multiple choice
4.) What is the unit for density?
a)kg/m3
b)m3 per kg
c)Density unit
d)Nm/s
5.) When an acceleration is present, it needs to be caused
by
a)a counter acceleration
b)the work done
c)a force
d)friction
6.) Gravitational potential energy depends on
a)mass and height
b)mass and acceleration due to gravity
c)acceleration due to gravity, mass and height
d)none of the mentioned parameters
Multiple choice
4.) What is the unit for density?
a)kg/m3
b)m3 per kg
c)Density unit
d)Nm/s
5.) When an acceleration is present, it needs to be caused
by
a)a counter acceleration
b)the work done
c)a force
d)friction
6.) Gravitational potential energy depends on
a)mass and height
b)mass and acceleration due to gravity
c)acceleration due to gravity, mass and height
d)none of the mentioned parameters
Multiple choice
7.) For a stone thrown horizontally from a building with height h, the
horizontal component can be treated
a)the same as the horizontal
b)as accelerated motion, but the horizontal part has to be accounted for
c)independently from the horizontal and has no acceleration
d)independently from the horizontal and has acceleration
8.) An empty and full cylinder made from the same material are rolling down
an incline. The solid cylinder is
a)slower
b)same speed
c)faster
d)twice as fast
9.) Angular momentum depends on angular velocity
a)quadratic
b)linear
c)not at all
d)linear but like 1/x
10.) A feather and a stone released from the top of a building will hit the
ground
a)at the same time
b)the stone will be earlier
c)the feather will be earlier
d)it can be either way
Multiple choice
7.) For a stone thrown horizontally from a building with height h, the
horizontal component can be treated
a)the same as the horizontal
b)as accelerated motion, but the horizontal part has to be accounted for
c)independently from the horizontal and has no acceleration
d)independently from the horizontal and has acceleration
8.) An empty and full cylinder made from the same material are rolling down
an incline. The solid cylinder is
a)slower
b)same speed
c)faster
d)twice as fast
9.) Angular momentum depends on angular velocity
a)quadratic
b)linear
c)not at all
d)linear but like 1/x
10.) A feather and a stone released from the top of a building will hit the
ground
a)at the same time
b)the stone will be earlier
c)the feather will be earlier
d)it can be either way
Short questions
• To prepare for this part, you will have to
look at the lecture notes
• It is also good practice to use the textbook
• The descriptions are a little different,
therefore you can test if you really
understood what we were talking about
• These are to test your understanding of the
concept and require none (or very little
math skills)
11.) Formulate Newtonʼs second law in your own words!
12.) Draw a free body diagram of a floating raft.
13.) Describe kinetic energy in your own words (include
formula).
11.) Formulate Newtonʼs second law in your own words!
F = ma. Newtonʼs second law describes that the net force of a
system causes an acceleration of an object with mass m.
12.) Draw a free body diagram of a floating raft.
Buoyant force (upwards), Gravitational force downwards
Buoyant force needs to be larger.
13.) Describe kinetic energy in your own words (include
formula).
Energy of an object due to movement. It is related to the mass
of the object (linear) and the objects velocity (quadratic).
KE = 1/2mv2
14.) Explain why (and how) angular momentum is conserved for a spinning
skater with arms out compared to arm pulled close the the body. (2 points)
15.) In linear motion with constant acceleration the following equation is true:
vf2 = vi2 + 2ad with vi initial velocity, vf final velocity, a acceleration and d
displacement. How does this expression change for rotational motion - write
down the formula and name the parts as done above.
16.) How would you determine the density of an irregularly shaped rock?
(2 points)
Angular momentum is defined as L = iω. The skater with arms out will have a
lower angular velocity, but a larger moment of inertia than the skater with
arms in. The changes compliment each other, so that angular momentum
remains the same.
15.) In linear motion with constant acceleration the following equation is true:
vf2 = vi2 + 2ad with vi initial velocity, vf final velocity, a acceleration and d
displacement. How does this expression change for rotational motion - write
down the formula and name the parts as done above.
ωf2 = ωi2 + 2αθ
with ω angular velocity, α angular acceleration and Θ
angular displacement
16.) How would you determine the density of an irregularly shaped rock?
(2 points)
Use Archimedes principle: Fill a container of known volume with water to the
rim. Place object in container, collect water that spills and determine itʼs weight.
The volume of the displaced water can be used to determine the density of the
irregularly shaped rock.
17.) What do we mean, when we talk about the center of gravity - describe in
your own words.
18.) Describe what we mean, when we are talking about an inelastic collision?
19.) A bomb, initially at rest, explodes into several pieces. Is linear momentum of
the system conserved? Is kinetic energy of the system conserved? (2 points)
20.) Two thin-walled drinking glasses having equal base areas but different crosssectional areas above the base, are filled to the same level with water.
According to the expression P = P0 + ρgh, the pressure is the same at the
bottom of both glasses. Why does one glass weigh more than the other?
(2 points)
17.) What do we mean, when we talk about the center of gravity - describe in
your own words.
For an extended body this is the single point (balance point) on which the
gravitational force acts effectively - it depends on the distribution of the mass in
the body. (And is the same as the center of mass, when g is the same for the
whole body)
18.) Describe what we mean, when we are talking about an inelastic collision?
The shape of the involved objects will change. Kinetic energy is not conserved.
19.) A bomb, initially at rest, explodes into several pieces. Is linear momentum of
the system conserved? Is kinetic energy of the system conserved? (2 points)
The linear momentum is conserved. The velocities of each piece added as
vectors need to be zero.
Kinetic energy is conserved as well for the complete system. Assuming we can
neglect loss into heat and so on.
(Both answerʼs for kinetic energy counted, if explanation was added)
20.) Two thin-walled drinking glasses having equal base areas but different crosssectional areas above the base, are filled to the same level with water.
According to the expression P = P0 + ρgh, the pressure is the same at the
bottom of both glasses. Why does one glass weigh more than the other?
(2 points)
One has more weight, because it has larger volume, therefore more fluid and
therefore more mass. Since g is the same for both, more mass means more
weight.
21.) A 2.0 kg block of slippery cheese that slides along a frictionless track
from point a to point b. The cheese travels through a total distance of 2.0 m
along the track, and a net vertical distance of 0.80 m. How much work is
done on the cheese by the gravitational force?
Work done by the gravitational force acts vertical: W = F cosΘ d = mgd
Here: Θ = 0, therefore cos0 = 1 (same direction)
d given as 0.80 m
g = 9.8 m/s2
m = 2.0 kg
Therefore W = mgd = (2.0 kg)(9.8 m/s2)(0.80 m) = 15.7 Nm = 15.7 J
22.) (4 points) Calculate the force exerted on your eardrum due to the water
when you are swimming at the bottom of a pool that is 5.0 m deep. Assume
that the pressure of the air inside you ear is 1.0 x 105 Pa and the surface
area of the eardrum is 1 x 10-4 m2
The formula needed was given above #20: P = P0 + ρgh
Also need definition of pressure P = F/A
(force over area)
P = P0 + ρgh = 1.0 x 105 Pa + 1000 kg/m3 (9.8 m/s2)(5.0m) = 1.49 x 105 Pa
F = P x A = (1.49 x 105 Pa)(1.0 x 10-4 m2) = 14.9 N
UNITS:
Pa = N/m2 (so force calculation fine)
N = kgm/s2
Pressure calculation: kg /ms2 = N/m2 = Pa
23.) A cylinder is standing on one of itʼs sides, it has a radius of 5 cm.
How much pressure does the ground underneath experience?
P = F/A = mg/A = m {(9.8 m/s)/(π (0.05m)2) = m 1248 m/s2
depends on mass
24.) How is elastic potential energy defined? Describe the relations.
PE(elastic) = 1/2 k x2
where k is the spring constant - linear dependence
and x the amount the spring stretches or compresses - quadratic dependence
25.) To write down a formula for Happiness H, we know about the following relations:
o H is increasing linear when more laughter L is in your life
o H is inverse proportional to tragedies T
o H is proportional to “falling in love” square, where falling is love is symbolized by F
25.) To write down a formula for Happiness H, we know about the following relations:
o H is increasing linear when more laughter L is in your life
o H is inverse proportional to tragedies T
o H is proportional to “falling in love” square, where falling is love is symbolized by F
We need a proportionality constant, which we can call K and then we follow the recipe:
H = K L/T F2
Problems
• Remember, that the bulk of the points are
given for drawings, reasoning (= describing
concept and route to solution), formula’s to
use. The actual calculations are the last part
and give only a few points
• Be efficient and smart about solving them
• Make comments, if you are sure, that your
result is wrong and only try again, if you
have time left over at the end
A.) The end of a ski jump (15 points)
A ski jumper leaves the ski track moving in the horizontal direction with a speed of
25.0 m/s. The landing incline below him falls off with a slope of 35.0°.
(a)Where does she land on the incline? (10 points)
(b)Suppose everything in this example is the same except the ski jump is curved so
that the jumper is projected upward at an angle from the end of the track. Is this
design better in terms of maximizing the length of the jump? Explain your answer.
(5 points)
(a) We have an initial velocity in x direction of 25 m/s, initial velocity in y direction is zero.
o We define our origin at the point where the jumper leaves the slope.
o We are looking for horizontal distance x and vertical distance y (negative) to find the
coordinates of the landing point.
o Horizontal and vertical part can be treated separately. Horizontal part has constant
velocity, so x(final) = v(x initial) t
o Vertical part is constant acceleration (free fall), so use
y(final) = y (initial) t + 1/2 a t2
(b) When the slope is curved upwards, the initial velocity in y direction is not zero
anymore. Therefore the jumper spends more time in the air. This can lead to the fact
that the jumper gets further.
But let’s look at the extreme of the slope being bent upwards by 90 degrees, than the
jumper would have no horizontal component, go straight up and fall down at the same
point.
Therefore there is an ideal angle (for every given set of parameters), that maximizes
the distance a jumper can go.
RESULTS: x(final) = 89.3 m
y(final) = -62.5 m
and Φ = 53°
B.) Letʼs go the Loop-the-Loop! (15 points)
A pilot of mass m in a jet aircraft executes a loop-the-loop. In this maneuver, the aircraft
moves in a vertical circle of radius 2.70 km at a constant speed of 225 m/s.
a)Make a drawing of the situation and indicate the velocity vector at the top and the
bottom of the circle
(1 point)
b)Draw two free body diagrams for the pilot - one on the top of the circle and one on the
bottom
(4 points)
c)Determine the area of the circle and its circumference (2 points)
d)Determine the force exerted by the seat on the pilot at the bottom of the loop. Express
your answer in terms of the weight of the pilot. (5 points)
e)Determine the force exerted by the seat on the pilot at the top of the loop.
(3 points)
(a) circle, radius, velocity tangential - realize this is uniform circular motion
(b) bottom: normal force up, gravitational force down (smaller)
top: both forces act downward, gravitational force larger
(c) area of circle: A = πr2 = π (2700 m)2 = 2.29 x 107 m2
circumference circle: C = 2πr = 2π (2700 m) = 16965 m = 1.70 x 104 m
(d) Newton’s second law: ΣF = FN(bottom) - mg = maC = m v2/r
[Result: 2.91 mg]
(e) Newton’s second law: ΣF = FN(top) - mg = maC = m v2/r
[Result: 0.913 mg]
C.) The Spring-Loaded Popgun (15 points)
The launching mechanism of a popgun consists of a spring of unknown spring
constant. The spring is compressed by 0.120 m, when the gun is fired vertically (up),
it is able to launch a 35.0 g projectile to a maximum height of 20.0 m above the
position of the projectile as it leaves the spring.
a)Neglecting all resistive forces, determine the spring constant (8 points)
b)Find the speed of the projectile as it moves through the equilibrium position of the
spring, just before it leaves the spring. (7 points)
(a) Conservation of mechanical energy is the concept to use for this problem:
KE(0) + PEG(0) + PEE(0) = KE(f) + PEG(f) + PEE(f)
Left side: no kinetic energy, no potential energy for relaxed position
no kinetic energy after release (right side either)
So: mgy(0) = mgy(f) +1/2 kx2
Solve for k: k = 2mg (y(0) - y(f))/x2
Now: m = 0.035 kg // g = 9.8 m/s2 // y(0) = 20 m // y(f) = -0.120 m
[Result k=958 N/m]
(b) Same concept only now the left side has kinetic energy
1/2 mv2 = mgy + 1/2 kx2 Solve for v ...
m, g, x as before, k we just calculated, y = y(f)
[Result: v = 19.8 m/s]
D.) Proton-Proton Collision
(15 points)
A proton collides elastically with another proton that is initially at rest. The
incoming proton has an initial speed of 3.50 x 105 m/s and makes a glancing
collision with the second proton. After the collision, one proton moves off at an
angle of 37° to the original direction of motion and the second deflects at an angle
of Φ to the same axis.
a)Make a drawing of the collision (before, during, after) and include all important
parameters (4 points)
b)Find the final speed of the two protons (after the collision) (8 points)
c)Determine the angle Φ (3 points)
This problem needs momentum conservation to be solved, like usually for collision
problems (similar to pool problem we had in class)
The calculations here are a little tricky, but it should be no problem to write down the
conservation in horizontal and vertical components:
m1 = m2 (therefore can be eliminated algebraically)
horizontal: v(1,0) = v(1,f) cosΘ + v(2,f) cosϕ
(1)
vertical: 0 = v(1,f) sinΘ - v(2,f) sinΦ
(2)
This gives us two equations. But we are looking for 3 parameters {v(1,f), v(2,f) and Φ}, so
we need a third equation, which we get by using energy conservation:
v(1,f)2 + v(2,f)2 = v(1,0)2 + v(2,0)2
with v(2,0) = 0 (at rest)
Results: v(1,f) = 2.80 x 105 m/s
v(2,f) = 2.11 x 105 m/s
Φ = 53°