Question 1 [ Work ]: A constant force, F, is applied to a block of mass m on an inclined plane as
shown in Figure 1. The block is moved with a constant velocity by a distance s. The coefficient
of kinetic friction between the inclined plane and the block is μk.
(a) Draw free-body diagrams and find the force F (5 points).
(b) What is the net work done by F (10 points)?
(c) Check case (a) if α=0 and comment on the
result (5 points).
Figure 1
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Solution 1:
(a) From Newton’s 2nd law:
∑
cos
∑
sin
0, y 0 and
x
sin
cos
F
(1)
(2)
α
(3)
·
sin
1
(b)
0
tan
sin
fk
m
mg
Free-body diagram
cos
sin
cos
N
θ
From (1), (2), and (3)
sin
cos
y
̂
sin
̂ ·
̂
cos
cos
tan
0 then
cos
The first term is the work done against the gravitation, the second term is the work done
against friction.
x
Question 2 [ Conservation of Energy ]: A block of mass m is released from an inclined plane as
shown in Figure 2. The coefficient of kinetic friction between the inclined plane and the block is
μk. Assume that the spring is initially uncompressed and there is no friction on the horizontal
surface.
(a) Find the maximum compression in the spring (10 points).
(b) If the block is bounced back from the spring, find the maximum height on the inclined
plane that the block can reach (7 points).
(c) Check case (a) if μk =0 and comment on the result (3 points).
Figure 2
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Solution 2:
(a) The total initial energy
sin
(1)
The speed of the block when it reaches the bottom of
the inclined plane:
1
sin
cos
2
1
2
2
sin
(c) If
0, then
is conserved).
m
θ
mg
fk
cos
(b) When the block reaches at maximum height on the inclined plane,
1
cos
2
sin
sin
1
x
N
Free-body diagram
The maximum compression in the spring
1
2
y
0
cos
cot
sin , the block is returned back its initial height (Total energy
Question 3 [ Momentum ]: A man of mass m clings to a rope ladder
suspended below a balloon of mass M as shown in Figure 3. The balloon
is stationary with respect to the ground .
(a) If the man begins to climb the ladder at a speed v (with respect to
the ladder), in what direction and with what speed (with respect to
Earth) will the balloon move? (12 points).
(b) If the man then stops climbing, what is the speed of the balloon
(with respect to Earth) (4 points)?
(c) Assuming that M >> m, what is the speed of the balloon in case
(a) (4 points)?
Figure 3
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Solution 3:
Method 1: Center of mass
(a) The net force on the balloon-man system is zero so the center of mass of this system is
constant:
cm
cm
0
The balloon is moving downward.
(b) If
0
0, the balloon is stationary.
/
(c) If
0,
0 : expected
Method 2: Conservation of momentum
(a) The momentum is conserved:
0,
,
Question 4 [ Collision ]: The two balls, m2 and m3, on the right of Figure 4 are slightly separated
and initially are at rest; the left ball, m1, is incident on m2 with speed v0. Assuming head-on
elastic collisions and no friction;
(a) Find speeds of m1 and m2 just after
v0
the first collision (8 points).
(b) Assuming that m1=m2=m and m3 ≤ m,
m1
m2 m3
show that there are two collisions and
find all final velocities (6 points).
(c) Assuming that m1=m2=m and m3 > m,
Figure 4
show that there are three collisions and
find all final velocities (6 points).
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Solution 4:
(a) Due to elastic collision, momentum and energy are conserved:
,
1
1
1
,
2
2
2
(1)
(2)
From (1) and (2)
3
2
(b)
4
,
After first collision between
and
,
0 and
For the second collision, between
and
, we do not need to solve the equations
again, we can write directly from (3) and (4)
2
Since
,
collisions.
(c) Similarly, since
three collisions.
0
2
never returns back and collides with
,
returns back and collides with
, so we have only two
. In this case, we have
Question 5 [ Angular Motion ]: We could measure
the speed of light, c, by using a rotating slotted
wheel. As shown in the Figure 5, a beam of light
passes through one of the slots at the outside edge
of the wheel, travels to a mirror, and returns to the
wheel just in time to pass through the next slot in
the wheel. Assuming that the angular velocity of
wheel is w, number of slots around wheel’s edge is
N, and the distance between the wheel and the
mirror is L, find the speed of light c (20 points).
Figure 5
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Solution 5:
In the time light takes to go from the wheel to the mirror and back again, the wheel turns through
an angle
2
∆
The time is
∆
So the angular velocity of the wheel is
∆
∆
2