PHYSICS C
SECTION I, MECHANICS
Time—45 minutes
35 Questions
Directions: Each of the questions or incomplete statements below is followed by
five suggested answers or completions. Select the one that is best in each case and
then mark it on your answer sheet.
1. A rock is dropped off a cliff and falls the first half of the distance to the
ground in t1 seconds. If it falls the second half of the distance in t2
seconds, what is the value of t2/t1? (Ignore air resistance.)
(A) 1/(2 )
(B) 1/
(C) 1/2
(D) 1 −(1/ )
(E) −1
2. A bubble starting at the bottom of a soda bottle experiences constant
acceleration, a, as it rises to the top of the bottle in some time, t. How
much farther does it travel in the last second of its journey than in the
first second? Assume that the journey takes longer than 2 seconds.
(A) a(t + 1 s)2
(B) a(t – 1 s)2
(C) at2
(D) a(t + 1 s)(1 s)
(E) a(t – 1 s)(1 s)
3. An object initially at rest experiences a time-varying acceleration given
by a = (2 m/s3)t for t ≥ 0. How far does the object travel in the first 3
seconds?
(A) 9m
(B) 12 m
(C) 18 m
(D) 24 m
(E) 27 m
4. What is the fewest number of the following conditions to ensure that
angular momentum is conserved?
I. Conservation of linear momentum
II. Zero net external force
III. Zero net external torque
(A) II only
(B) III only
(C) I and II only
(D) I and III only
(E) II and III only
5. In the figure shown, a tension force FT causes a particle of mass m to
move with constant angular speed ω in a horizontal circular path (in a
plane perpendicular to the page) of radius R. Which of the following
expressions gives the magnitude of FT? Ignore air resistance.
(A) mω2R
(B)
(C)
(D) m(ω2R − g)
(E) m(ω2R + g)
6. An object (mass = m) above the surface of the Moon (mass = M) is
dropped from an altitude h equal to the Moon’s radius (R). With what
speed will the object strike the lunar surface?
(A)
(B)
(C)
(D)
(E)
7. Pretend that someone managed to dig a hole straight through the
center of the Earth all the way to the other side. If an object were
dropped down that hole, which of the following would best describe its
motion? Assume ideal conditions (Earth is a perfect sphere, there are
no dissipative forces) and that the object cannot be destroyed.
(A) It would fall to the center of the Earth and stop there.
(B) It would fall through the hole to the other side, continue past the
opposite side’s opening, and fly into space.
(C) It would oscillate back and forth from one opening to the other
indefinitely.
(D) It would oscillate back and forth, but the amplitude would
decrease each time, eventually settling at the center of the
Earth.
(E) It would fall to the other side and stop there.
8. A uniform cylinder of mass m and radius r unrolls without slipping
from two strings tied to a vertical support. If the rotational inertia of
the cylinder is mr2, find the acceleration of its center of mass.
(A) g
(B) g
(C) g
(D) g
(E) g
9. A uniform cylinder, initially at rest on a frictionless, horizontal surface,
is pulled by a constant force F from time t = 0 to time t = T. From time
t = T on, this force is removed. Which of the following graphs best
illustrates the speed, v, of the cylinder’s center of mass from t = 0 to t =
2T?
(A)
(B)
(C)
(D)
(E)
10. A space shuttle is launched from Earth. As it travels up, it moves at a
constant velocity of 150 m/s straight up. If its engines provide 1.5 × 108
W of power, what is the shuttle’s mass? You may assume that the
shuttle’s mass and the acceleration due to gravity are constant.
(A) 6.7 × 102 kg
(B) 1.0 × 105 kg
(C) 6.7 × 105 kg
(D) 1.0 × 106 kg
(E) 2.3 × 106 kg
11. A satellite is in circular orbit around Earth. If the work required to lift
the satellite to its orbit height is equal to the satellite’s kinetic energy
while in this orbit, how high above the surface of Earth (radius = R) is
the satellite?
(A) R
(B) R
(C) R
(D) R
(E) 2R
12. The figure above shows a uniform bar of mass M resting on two
supports. A block of mass M is placed on the bar twice as far from
Support 2 as from Support 1. If F1 and F2 denote the downward forces
on Support 1 and Support 2, respectively, what is the value of F2/F1?
(A)
(B)
(C)
(D)
(E)
13. A rubber ball (mass = 0.08 kg) is dropped from a height of 3.2 m and,
after bouncing off the floor, rises almost to its original height. If the
impact time with the floor is measured to be 0.04 s, what average force
did the floor exert on the ball?
(A) 0.16 N
(B) 16 N
(C) 32 N
(D) 36 N
(E) 64 N
14. A disk of radius 0.1 m initially at rest undergoes an angular
acceleration of 2.0 rad/s2. If the disk only rotates, find the total
distance traveled by a point on the rim of the disk in 4.0 s.
(A) 0.4 m
(B) 0.8 m
(C) 1.2 m
(D) 1.6 m
(E) 2.0 m
15. In the figure above, a small object slides down a frictionless quarter-
circular slide of radius R. If the object starts from rest at a height equal
to 2R above a horizontal surface, find its horizontal displacement, x, at
the moment it strikes the surface.
(A) 2R
(B) R
(C) 3R
(D) R
(E) 4R
16. The figure above shows a particle executing uniform circular motion in
a circle of radius R. Light sources (not shown) cause shadows of the
particle to be projected onto two mutually perpendicular screens. The
positive directions for x and y along the screens are denoted by the
arrows. When the shadow on Screen 1 is at position x = –(0.5)R and
moving in the +x direction, what is true about the position and velocity
of the shadow on Screen 2 at that same instant?
(A) y = –(0.866)R; velocity in –y direction
(B) y = –(0.866)R; velocity in +y direction
(C) y = –(0.5)R; velocity in –y direction
(D) y = +(0.866)R; velocity in –y direction
(E) y = +(0.866)R; velocity in +y direction
17. The figure shows a view from above of two objects attached to the end
of a rigid massless rod at rest on a frictionless table. When a force F is
applied as shown, the resulting rotational acceleration of the rod about
its center of mass is kF/(mL). What is k?
(A)
(B)
(C)
(D)
(E)
18. A toy car and a toy truck collide. If the toy truck’s mass is double the
toy car’s mass, then, compared to the acceleration of the truck, the
acceleration of the car during the collision will be
(A) double the magnitude and in the same direction
(B) double the magnitude and in the opposite direction
(C) half the magnitude and in the same direction
(D) half the magnitude and in the opposite direction
(E) dependent on the type of collision
19. A homogeneous bar is lying on a flat table. Besides the gravitational
and normal forces (which cancel), the bar is acted upon by exactly two
other external forces, F1 and F2, which are parallel to the surface of the
table. If the net force on the rod is zero, which one of the following is
also true?
(A) The net torque on the bar must also be zero.
(B) The bar cannot accelerate translationally or rotationally.
(C) The bar can accelerate translationally if F1 and F2 are not applied
at the same point.
(D) The net torque will be zero if F1 and F2 are applied at the same
point.
(E) None of the above
20. An astronaut lands on a planet whose mass and radius are each twice
that of Earth. If the astronaut weighs 800 N on Earth, how much will
he weigh on this planet?
(A) 200 N
(B) 400 N
(C) 800 N
(D) 1,600 N
(E) 3,200 N
21. A particle of mass m = 1.0 kg is acted upon by a variable force, F(x),
whose strength is given by the graph given above. If the particle’s speed
was zero at x = 0, what is its speed at x = 4 m?
(A) 5.0 m/s
(B) 8.7 m/s
(C) 10 m/s
(D) 14 m/s
(E) 20 m/s
22. The radius of a collapsing spinning star (assumed to be a uniform
sphere with a constant mass) decreases to of its initial value. What
is the ratio of the final rotational kinetic energy to the initial rotational
kinetic energy?
(A) 4
(B) 16
(C) 162
(D) 163
(E) 164
23. A ball is projected with an initial velocity of magnitude v0 = 40 m/s
toward a vertical wall as shown in the figure above. How long does the
ball take to reach the wall?
(A) 0.25 s
(B) 0.6 s
(C) 1.0 s
(D) 2.0 s
(E) 3.0 s
24. If C, M, L, and T represent the dimensions of charge, mass, length, and
time respectively, what are the dimensions of the permittivity of free
space (ɛ0)?
(A) T2C2/(M2L2)
(B) T2C2/(ML3)
(C) ML3/(T2C2)
(D) C2M/(T2L2)
(E) T2L2/(C2M)
25. The figure shown is a view from above of two clay balls moving toward
each other on a frictionless surface. They collide perfectly inelastically
at the indicated point and are observed to then move in the direction
indicated by the post-collision velocity vector, v’. If m1 = 2m2, and v’ is
parallel to the negative y-axis, what is v2?
(A) v1(sin 45°)/(2 sin 60°)
(B) v1(cos 45°)/(2 cos 60°)
(C) v1(2 cos 45°)/(cos 60°)
(D) v1(2 sin 45°)/(sin 60°)
(E) v1(cos 45°)/(2 sin 60°)
26. In the figure above, the coefficient of static friction between the two
blocks is 0.80. If the blocks oscillate with a frequency of 2.0 Hz, what is
the maximum amplitude of the oscillations if the small block is not to
slip on the large block?
(A) 3.1 cm
(B) 5.0 cm
(C) 6.2 cm
(D) 7.5 cm
(E) 9.4 cm
27. When two objects collide, the ratio of the relative speed after the
collision to the relative speed before the collision is called the
coefficient of restitution, e. If a ball is dropped from height H1 onto a
stationary floor, and the ball rebounds to height H2, what is the
coefficient of restitution of the collision?
(A) H2/H1
(B) H2/H1
(C)
(D)
(E) (H1/H2)2
28. The figure above shows a square metal plate of side length 40 cm and
uniform density, lying flat on a table. A force F of magnitude 10 N is
applied at one of the corners, as shown. Determine the torque
produced by F relative to the center of rotation.
(A) 0 N•m
(B) 1.0 N•m
(C) 1.4 N•m
(D) 2.0 N•m
(E) 4.0 N•m
29. A small block of mass m = 2.0 kg is pushed from the initial point (xi, zi)
= (0 m, 0 m) upward to the final point (xf, zf) = (3 m, 3 m) along the
path indicated. Path 1 is a portion of the parabola z = x2, and Path 2 is a
quarter circle whose equation is (x – 1)2 + (z – 3)2 = 4. How much work
is done by gravity during this displacement?
(A) −60 J
(B) −80 J
(C) −90 J
(D) −100 J
(E) −120 J
30. In the figure shown, the block (mass = m) is at rest at x = A. As it
moves back toward the wall due to the force exerted by the stretched
spring, it is also acted upon by a frictional force whose strength is given
by the expression bx, where b is a positive constant. What is the block’s
speed when it first passes through the equilibrium position (x = 0)?
(A) A
(B) A
(C) A
(D) A
(E) A
31. The rod shown above can pivot about the point x = 0 and rotates in a
plane perpendicular to the page. Its linear density, λ, increases with x
such that λ(x) = kx, where k is a positive constant. Determine the rod’s
moment of inertia in terms of its length, L, and its total mass, M.
(A) ML2
(B) ML2
(C) ML2
(D) ML2
(E) 2ML2
32. A particle is subjected to a conservative force whose potential energy
function is
U(x) = (x – 2)3 – 12x
where U is given in joules when x is measured in meters. Which of the
following represents a position of stable equilibrium?
(A) x = –4
(B) x = –2
(C) x=0
(D) x=2
(E) x=4
33. A light, frictionless pulley is suspended from a rigid rod attached to the
roof of an elevator car. Two masses, m and M (with M > m), are
suspended on either side of the pulley by a light, inextendable cord.
The elevator car is descending at a constant velocity. Determine the
acceleration of the masses.
(A) (M − m)g
(B) (M + m)g
(C)
(D)
(E) (M – m)(M + m)g
34. A particle’s kinetic energy is changing at a rate of –6.0 J/s when its
speed is 3.0 m/s. What is the magnitude of the force on the particle at
this moment?
(A) 0.5 N
(B) 2.0 N
(C) 4.5 N
(D) 9.0 N
(E) 18 N
35. An object of mass 2 kg is acted upon by three external forces, each of
magnitude 4 N. Which of the following could NOT be the resulting
acceleration of the object?
(A) 0 m/s2
(B) 2 m/s2
(C) 4 m/s2
(D) 6 m/s2
(E) 8 m/s2
STOP
END OF SECTION I, MECHANICS
PHYSICS C
SECTION II, MECHANICS
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes per
question for answering each of the questions, which are worth 15 points each. The
parts within a question may not have equal weight.
Mech 1. An ideal projectile is launched from the ground at an angle θ to the
horizontal, with an initial speed of v0. The ground is flat and level
everywhere. Write all answers in terms of v0, q, and fundamental
constants.
(a) Calculate the time the object is in the air.
(b) Calculate the maximum height the object reaches.
(c) What is the net vertical displacement of the object?
(d) Calculate the range (horizontal displacement) of the object.
(e) What should θ be so that the projectile’s range is equal to its
maximum vertical displacement?
Mech 2. A narrow tunnel is drilled through Earth (mass = M, radius = R),
connecting points P and Q, as shown in the diagram on the left below.
The perpendicular distance from Earth’s center, C, to the tunnel is x. A
package (mass = m) is dropped from Point P into the tunnel; its distance
from P is denoted y and its distance from C is denoted r. See the diagram
on the right.
(a) Assuming that Earth is a homogeneous sphere, the gravitational
force F on the package is due to m and the mass contained within
the sphere of radius r < R. Use this fact to show that
F=
(b) Use the equation F(r) = –dU/dr to find an expression for the
change in gravitational potential energy of the package as it
moves from Point P to a point where its distance from Earth’s
center is r. Write your answer in terms of G, M, m, R, and r.
(c) Apply Conservation of Energy to determine the speed of the
package in terms of G, M, R, x, and y. (Ignore friction.)
(d) (i) At what point in the tunnel—that is, for what value of y—will
the speed of the package be maximized?
(ii) What is this maximum speed? (Write your answer in terms
of G, M, R, and x.)
Mech 3. The diagram below is a view from above of three sticky hockey pucks on
a frictionless horizontal surface. The pucks with masses m and 2m are
connected by a massless rigid rod of length L and are initially at rest. The
puck of mass 3m is moving with velocity v directly toward puck m. When
puck 3m strikes puck m, the collision is perfectly inelastic.
(a) Immediately after the collision,
(i) where is the center of mass of the system?
(ii) what is the speed of the center of mass? (Write your answer
in terms of v.)
(iii) what is the angular speed of the system? (Write your answer
in terms of v and L.)
(b) What fraction of the system’s initial kinetic energy is lost as a
result of the collision?
STOP
END OF SECTION II, MECHANICS
