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Final Exam Solutions Key
Lise Wills
Phys 113 Contemporary Physics
Drexel University
9 December 2014
1. [25] Short Answer (5 points each)
(a) What are the mediating particle(s) for:
i. Electromagnetism
The photon, γ
ii. The Weak Force
W , Z
iii. The Strong Force
Gluons
(b) Using the moment of inertia information in the equation sheet, sort the speed in which the
following objects would roll down an inclined plane from fastest to slowest:
•
•
•
•
A
A
A
A
solid sphere
solid disk
ring
hollow sphere
Things with a smaller moment of inertial roll faster, so arrange the items by smallest moment of
inertia to largest, and for the one which do not have an equation on your sheet, you can extrapolate
from the disk/ring example.
1.
2.
3.
4.
Sphere: Isphere 25 M R2
Disk: Idisk 12 M R2
Shell: Ishell 23 M R2 +1 if correct since it wasn’t on the sheet, no penalty for incorrect.
Ring: Iring M R2
(c) A 0.2 kg ball collides elastically head-on with a 0.1 kg ball initially at rest at 30 m/s. How fast,
and in what direction, will the lighter ball move after the collision?
Please ignore rolling.
1
p2f
p1i
2m2
m1 m2
Since p2i
m2 v2f
v2f
v2f
m1 v1i
0,
p2i
m2 m1
m2 m1
2m2
m1 m2
m1
m v1i m 2m2m
2
1
2
0.2 kg
m
0.1 kg 30 s 2 0.30.2kgkg
40 ms
(d) Suppose, instead, that the 0.2 kg ball and the 0.1 kg ball were made of clay and stuck after
collision. How quickly, and in what direction, would the combined mass move after the collision?
Elastic collisions are easier as we only have to worry about conservation of momentum.
pf
vi mi vf mf
m
30 0.2 kg vf pm1
s
pi
m2 q
vf
0.2 kg
0.3
30 ms
kg
vf
20 ms
(e) A mono-energetic beam of electrons shines upon a sheet with two small slits cut out of it. Some
distance behind the sheet is a fluorescent screen which lights up when hit by an electron. Sketch
the fringe pattern that the electrons would have if they exhibit a) Classical Behavior, or b)
Quantum behavior.
Describe at least one way of forcing classical behavior.
a) Classical
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b) Quantum
It is also a fair representation to draw:
c) Forcing classical behavior:
We can force classical behavior by observing the quantum system as it passes through the slits.
2. [15] A 0.1kg puck starts at rest at the top of a plane with a coefficient of kinetic friction, µk 0.2
and a coefficient of static friction, µs 0.3 as shown. The plane is 10m long, and makes an angle of
75 degrees with respect to the horizontal.
3
(a) Please draw a free body diagram of the puck on the plane. Be sure to break your forces into
components parallel and perpendicular to the plane.
(b) How fast will the puck move at the bottom of the plane?
4
Ff r Wx
µmg cos θ mg sin θ
gmpµ cos θ sin θq
9.8 sm2 p0.1 kgqp0.2 cos 75 sin 75q
0.89N
W F d
0.89 N 10 m
8.9 J
∆KE
12 mv2
ΣFx
v
c
d
v
2W
m
2 8.9 J
0.1 kg
14 ms
(c) How much energy goes into heat?
W Q
m
mgh 0.1 kg 9.8 2 10 sin θ
s
9.5 J 8.9 J Q
Q 0.6 J
∆E
(d) E.C. Over what range of angles will the puck slide down the plane at all?
Upper limit is clearly 900 as then we have a verticle wall, and it will definitely fall down then. The
angle for the lower bound is the one just above when ΣFx 0 Since it’s not moving in this case,
µ 0.3.
µmg cos θ
mg sin θ
µ cos θ sin θ
µ tan θ
θ tan1 µ
16.70
16.70 θ ¤ 900
ΣFx
mg sin θ
µmg cos θ
3. [25] A spaceship has a mass of 10, 000kg, and a length of exactly 106 ls (300 m). It starts near earth
from rest, and is accelerated by applying 9 1020 J of work.
5
Note that you will need to do a number of calculations in this problem which rely on knowing the
speed/gamma factor of the ship. If you are unable to compute part a), please assume for concreteness
that γ 4 (not the correct answer).
(a) What is the γ factor on the ship after it has been accelerated?
9 10
∆KE
J mc2 pγ 1q
9 1020 J
γ
2
10, 000 kg 3 108 m
s
γ2
W
20
1
(b) What is the speed of the ship after it has been accelerated? Please express your answer as a
decimal fraction of the speed of light. Reminder: This is a relativistic problem. Think carefully.
γ
b
1
1
c
12 1
v2
c
v 2
c
v 2
c
1 41
c
c 1 14
v 0.86c
v
(c) Within the ship (the primed frame) a crewman throws a wrench toward the front of the ship at a
speed of 0.5c. Draw a spacetime diagram of the two events: throwing the wrench, and it hitting
the front of the ship, as seen from the perspective of the ship’s crew. For convenience, you may
set the throw at t1 0, and x1 0. Include also worldlines representing the back and front of the
ship.
Anything going at 0.5c has a line of y x2 .
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You can plot these or solve these on your calculator to help you get an idea of the values:
(d) Draw a spacetime diagram of the same two events as seen from earth. Include also worldlines
representing the back and front of the ship.
(e) How fast is the wrench moving as seen from earth?
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u1 v
1 uc2v
0.5c 0.86c
u
1
1
p0.5cqp0.86cq
c2
1 1.36c
0.43
u 0.95c
4. [20] Below, please find the potential energy diagram for gravity near mercury, set for units for a 1000
kg spaceship. The minimum radius plotted corresponds to the surface of mercury. For reference:
Rmercury
2.44 106 m
(a) The rocket starts from the surface of mercury and applies a short thrust, yielding 6 1010 J of
kinetic energy. Does the rocket escape Mercury’s gravity, and if not, how high above the surface
does it get before plummeting down?
This energy is an order of magnitude larger than the potential energy of Mercury. The spaceship
escapes.
(b) What is the escape velocity of the rocket from the surface of mercury?
At the surface of Mercury, U 9 109 J. To escape fully, we need to completely convert this
potential energy to kinetic energy.
8
KE
1
2
9 109 J mvesc
2
d
2p9 109 J q
v U
esc
vesc
1000 kg
4262.64 ms
(c) Based on your previous answer (and the given radius above), what is the mass of mercury?
GMr m
U r
M
U
M
Gm
9 109 J 2.44 106 m
3
6.67 1011 kgms2 1000 kg
3.29 1023 kg
(d) Estimating from the plot above (using rise over run), estimate the gravitational force on the rocket
near the surface of mercury. From that, please estimate the surface gravity. Note: I am looking
for you to use the plot to compute the gravitational force. I recognize that you have a general
equation to compute gravitational force and acceleration (which you may want to apply to check
your answers), but for now I am looking to see that you know how to read energy diagrams.
Estimating from the graph - we’re generous so long as you’re in the right ballpark!
F
F
dU
dr
p7 9q 109 J
p3.5
2.4q 106 m
2 109 J
1.1.
106 m
1.82 103 N
So we can check our answers, I will calculate gm .
gm
F
m
10 N
1.451000
kg
3
gm
1.8 sm2
Checking:
9
g
GM
r2
g
6.67 1011 kgms2
p3.29 1023 kgq
p2.44 106 mq2
3
3.68 sm2
Same order of magnitude is good enough for reading off of a graph in this circumstance. Errors
at least this large will be accepted.
5. [15] A quantized energy system has 3 possible states:
• n 1 ; 1 eV
• n 2; 4 eV
• n 3; 9 eV
Incidentally, this are simply the energy levels for a particle confined in a 1d box.
(a) List all possible energy transitions which give rise to an emitted photon. Including the upper and
lower level, and what the energy of the emitted photon would have.
∆E
• E3
Ñ E1 :
|Ef Ei |
|1 eV 9 eV |
∆E3Ñ1 8 eV
∆E3Ñ1
• E3
Ñ E2 :
|4 eV 9 eV |
∆E3Ñ1 5 eV
∆E3Ñ2
• E2
Ñ E1 :
|1 eV 4 eV |
∆E3Ñ1 3 eV
∆E2Ñ1
(b) What is the wavelength corresponding to a photon emitted for a 2 Ñ 1 transition? Please express
your answer in nm. What part of the spectrum/color is the photon?
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E
hc
λ
λ
hc
E
p
1034 Jsqp3 108 q ms
6.626
19 J q
p∆E eV qp1.6 10
1.242 106 m E1
1
6
1.242 10 m 3 eV
4.14 107 m
λ 414 nm Ñ Blue/purple
(c) Consider an incident photon with an energy of 2 eV that passes by a large group of these “particles
in a box” (some in each energy level). What happens to the photon?
This energy does not match the energy of any of the transitions above, so it passes through unhindered.
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