PHYS 172: Modern Mechanics
Spring 2010
Final exam
Comprehensive: all lectures, all homeworks
20 multiple choice questions (similar to midterm exams 1-3)
Equation sheet: will be attached
Lecture 27 – Review
Read 1.1 - 12.8
Newton’s first law of motion
An object moves in a straight line and at constant speed
except to the extent that it interacts with other objects
Inertial reference frame
Inertial frame: a frame in which Newton’s first law of motion is valid
Physical laws work in the same way in any inertial frame
Clicker
If a solid object rotates at constant angular velocity, is it possible to
chose an inertial reference frame in which the object will not rotate?
A) YES
B) NO
The Momentum Principle
The Momentum Principle
dp
= Fnet
dt
The rate of change of momentum is equal to
the net force acting on an object
p = γ mv
1
γ=
1−
v
c
2
The Momentum Principle
F1
dp
= Fnet
dt
F2
You and your friend push a box in opposite directions with
forces and as a result it moves at constant velocity to the
right. What is the net force applied to the system consisting
of a box?
A) Fnet = F1
B)
C)
D)
E)
Fnet = F2
Fnet = F1 + F2
Fnet = F1 − F2
None of the above
Clicker: Rate of change of momentum
dp
= Fnet
dt
Highest point
p=0
Equilibrium
Fspring
p
Lowest point
Fgrav
p=0
Fspring
p
Fgrav
1. What is
dp dt
at the equilibrium point?
2. What is
dp dt
at the lowest point?
3. What is
dp dt
at the highest point?
A) dp dt is up and maximized
B) dp dt = 0
C) dp dt is down and maximized
Clicker: Ferris wheel
Assume constant v
What forces act on a person?
gravity seat
6. Is the gravitational force constant? A) Yes
B) No
A) Yes
B) No
7. Is the net force |Fnet| constant?
Fnet = Fnet ,⊥ = mg + Fseat
8. Is |Fseat| constant?
Fseat
A) Yes
B) No
Fseat = Fnet − mg
Fnet ,⊥
mg
dp
dt
Weightless: what does it mean?
dp
dt
dp
= mg
dt
=p
⊥
=
||
v
= F⊥
R
dp
= F||
dt
Clicker 5: Turning car
A
D
B
C
What is the direction of
the net force acting on
people in the car?
Constant Gravitational Field
Clicker question:
g = 9.8 N/kg (or m/s 2 )
Due to the gravity, which components
of the velocity will change?
x, y and z
B) only x and y
C)only x
D)only y
E) only z
y
F = 0, −mg , 0
x
z
dp
= Fnet
dt
p fx = pix + Fnet , x ∆t
p fy = piy + Fnet , y ∆t
p fz = piz + Fnet , z ∆t
Clicker 7,8: truck and mosquito collide
Which experiences a larger ∆v ?
Which experiences a larger ∆p ?
A) Mosquito
B) Truck
C) The same
A) Mosquito
B) Truck
C) The same
During such an interaction, the changes in momenta are
equal and opposite, but not changes in velocity.
dp
= Fnet
dt
Reciprocity
F1on 2 = − F2 on1
Reciprocity (Newton’s 3rd law):
The forces of two objects on each other
are always equal and are directed in
opposite directions
∆p1 = F2on1∆t
∆p2 = F1on 2 ∆t
Momentum conservation:
∆p1 = −∆p2
∆psystem + ∆psurrounding = 0
Momentum and Energy
A
B
1. Which of the blocks acquired larger speed?
2. Which of the blocks acquired larger momentum?
A) Block A
B) Block B
C) equal
3. Which of the blocks acquired larger kinetic energy?
Momentum is proportional to time, not distance!
Work is proportional to the distance, but not time!
∆p = Fnet ∆t
W ≡ F|| ∆r = F ⋅ ∆r
Clicker question 5-6
5. The work done by the engine on accelerating car is
A) Negative
B) Zero
C)Positive
6. The work done by the brakes on a slowing car is
A) Negative
B) Zero
C)Positive
WF = F ⋅ ∆r = F ∆r cos θ
Clicker question 4
Moon moves along circular orbit around Earth.
The work done by Earth on Moon is:
A) Negative
B) Zero
C)Positive
WF = F ⋅ ∆r = F ∆r cos θ
Example: work done by a spring
equilibrium
f
W = F ⋅ dr
i
Fx = −kS x
x'
W = −kS xdx
F
0
x
0
x’
kS x 2
W =−
2
x'
0
k S x '2
=−
2
CLICKER:
1. What is the sign of work done by a spring when you compress it?
2. What is the sign of work done by a spring when you stretch it?
1
s
Clicker question
2
h
An object of mass m was pushed over the hill
of height h from point 1 to point 2 as shown.
The trajectory path length from point 1 to 2 is s.
How much work was done by the gravitational force?
A.
B.
C.
D.
E.
0J
mgh
2mgh
mgh/2
mgs
A) Positive
B) Negative
Potential Energy
r
A
B
Which of these diagrams represents repulsion?
Clicker: Energy graphs and types of orbits
Select energy graph that represents
the following motion:
A
B
C
The Energy Principle
∆Esystem = Wsurr + Q
“effect”
“cause” (due to interactions)
Conservation of Energy:
∆Esystem + ∆Esurroundings = 0
The rest energy, kinetic energy and momentum
E = γ mc =
mc 2
2
1 − v2 c2
Assume v = 0:
Rest energy of a particle
Erest = mc
2
E = mc 2 + K
Kinetic energy:
K ≡ E − mc 2
K=
mc 2
1− v c
2
p = γ mv
2
− mc 2
E = γ mc 2
Energy and momentum
Note: these are true for any speed!
E 2 − ( pc ) = ( mc 2 )
2
2
Two ways of thinking
1. The energy of a multiparticle system
consists of the individual particle energies
plus their pair-wise interactions
2. A system itself has energy, like a single
particle, and if the system is at rest (not
individual objects within the system!) its
energy E = Mc2, where M is mass of the
system.
i>Clicker question 1
O2 molecule is oscillating:
Mass of the molecule is:
A) Larger than total mass of two noninteracting oxygen atoms
B) Smaller than total mass of two noninteracting oxygen atoms
C) The same as mass of two noninteracting oxygen atoms
M = ( m1 + m2 + ...) +
K1 + K 2 + ... + U
c2
i>Clicker question 2
O2 molecule is oscillating:
Mass of the molecule is:
A) Larger when atoms are further apart
B) Larger when atoms are closer to each other
C) Larger when atoms are moving faster
D) It does not change during oscillation
M = ( m1 + m2 + ...) +
K1 + K 2 + ... + U
c2
Clicker
1. What happens with the mass of an object when
you heat it up?
A) It increases
B) It decreases
C)It does not change
i>clicker: hockey pucks
dPtot
= Fnet ,ext
dt
∆Ptot = FT ∆t
A
∆Ptot = FT ∆t
B
Two hockey pucks are pulled using attached strings as shown using
the same force. The string in case B is wrapped around the puck and
can unwind freely. In case A it is attached to the center.
Which hockey puck will accelerate quicker?
dPtot
= Fnet ,ext
dt
A) – puck A
B) – puck B
C) – the same
Example: hockey pucks
∆Ptot = FT ∆t
A
∆Ptot = FT ∆t
B
W1 = FT d1
A
d1
B
W2 = FT d 2
d2
A) – puck A
Which puck will acquire larger kinetic energy? B) – puck B
C) – the same
Translational kinetic energy
Ktot = K trans + K rel
K rel = K vib + K rot
2
2
Translational kinetic energy:
MvCM
P
Ktrans =
= tot
(motion of center of mass)
2
2M
(nonrelativistic case)
Clicker:
A system is initially at rest and consists of a man with a bottle sitting on ice
(ignore friction). The man then throws the bottle away as shown.
The velocity of the center of mass vcm will be:
A) Zero
B) Directed to right
C) Directed to left
dPtot
= Fnet ,ext
dt
http://www.punchstock.com/asset_images/95652058
Translational kinetic energy
Ktot = K trans + K rel
K rel = K vib + K rot
2
Translational kinetic energy:
MvCM
Ktrans =
(motion of center of mass)
2
(nonrelativistic case)
Clicker:
A system is initially at rest and consists of a man with a bottle sitting on ice
(ignore friction). The man then throws a bottle away as shown.
The translational kinetic energy of the point-particle
system will be:
A) Zero
B) > 0
C) < 0
http://www.punchstock.com/asset_images/95652058
Clicker question 1
1
Wheel 1 of mass M rolls down from a slope.
Wheel 2 of the same mass M slides down
from the same slope (ignore friction)
Which wheel will acquire larger total kinetic
energy?
2
A) Wheel 1 (rolling)
B) Wheel 2 (sliding)
C) The same
Clicker question 2
1
Wheel 1 of mass M rolls down from a slope.
Wheel 2 of the same mass M slides down
from the same slope (ignore friction)
Which wheel will acquire larger translational
kinetic energy?
2
A) Wheel 1 (rolling)
B) Wheel 2 (sliding)
C) The same
Clicker question 3
1
Wheel 1 of mass M rolls down a slope.
Wheel 2 of the same mass M slides down
the same slope (ignore friction)
Which of the wheels will get down first?
A) Wheel 1 (rolling)
B) Wheel 2 (sliding)
C) Both will get down in the same
time
2
A head-on collision: elastic
Ping-pong ball
p1
Tennis ball
p2 = 0
p4
p3
x
Momentum conservation:
Energy conservation:
p1x = p3 x + p4 x
p12x p32x p42x
=
+
2m 2m 2 M
Two equations
Two unknown
p3 x =
m±M
p1x
m+M
+ no interaction
- interaction
Example: head-on car crash
Two identical cars of mass m=1000 kg move toward each other with
speeds v1=v2=30 m/s (~65 mph), collide head-on and stick together.
CLICKER: What is the speed of the center of mass of the system
consisting of these two cars after the collision?
A.
B.
C.
D.
30 m/s
15 m/s
60 m/s
0 m/s
Example: head-on car crash
Two identical cars of mass m=1000 kg move toward each other with
speeds v1=v2=30 m/s (~65 mph), collide head-on and stick together.
CLICKER: What was the speed of the center of mass of the system
consisting of these two cars before the collision?
A.
B.
C.
D.
30 m/s
15 m/s
60 m/s
0 m/s
Example: head-on car crash
Two identical cars of mass m=1000 kg move toward each other with
speeds v1=v2=30 m/s (~65 mph), collide head-on and stick together.
CLICKER: During the collision, the total kinetic energy of the
system consisting of these two cars:
A. is conserved
B. increases
C. decreases
Clicker
LA = Ltrans , A + Lrot
Ltrans = rcm × Ptot
System: disk
Lrot = r1 × p1 + r2 × p2 + ...
L
=?
trans
In respect to A
A
Lrot = ?
A)
B)
C)
D)
E)
Zero
Out of page
Into the page
Up
Down
A)
B)
C)
D)
E)
Zero
Out of page
Into the page
Up
Down
torque :
Angular momentum dLA
=τA
principle dt
τ A ≡ rA × Fnet
Torque
A
τ A = rA Fnet sin θ
τA
B
rA⊥
C
τ A = rA⊥ Fnet
1. Which case corresponds to largest torque?
2. The direction of Torque in case B is:
A) into page
B) out of page
C) it has no direction since it is zero
A light comet moves around a very massive star along
elliptical orbit as shown.
B
C
D
A
Q: At which point the angular momentum of the comet
in respect to the position of the star maximizes?
A, B, C or D ?
E) Angular momentum does not change
dLA
=τA
dt
The fundamental assumption of statistical mechanics
Each microstate corresponding to a given
macrostate is equally probable.
Macrostate – total energy
Microstate – microscopic distribution of energy
The Second law of thermodynamics
If a closed system is not in equilibrium, the most
probable consequence is that the entropy will increase
Clicker 1
The entropies of two blocks made of
the same material are S1 and S2 and
depend on energy quanta q1 and q2 as
shown.
When brought into thermal contact,
which position on the graph
corresponds to equilibrium?
A)
B)
C)
D)
1 ∂S
=
T ∂E
A
B
C
D
S1 + S2
S1
S2
Clicker 2
A
Which of the blocks has higher
temperature in point C?
B
S1 + S2
A) The block described with S1
B) The block described with S2
S1
C) They have the same temperature
Clicker 1
The entropy as a function of internal energy for objects 1 and 2 is
shown in the graph
S
D
C
S2
1 ∂S
=
T ∂E
1
2
Assuming that both blocks have the
same internal energy E0, what can you
say about their temperatures?
A) T1 = T2
B) T1 > T2
C) T1 < T2
E0
E
Clicker 2
The entropy as a function of internal energy for objects 1 and 2 is
shown in the graph
S
1 ∂S
=
T ∂E
1
2
Assuming that block 2 internal energy
is E2, how much internal energy block 1
must have for its temperature to be
about the same?
A) E1
B) E2
C) E3
E1
E2
E3
E