AP PHYSICS C: MECHANICS
Scoring Guide
Unit 5 Progress Check: MCQ
1.
A sphere rotates about an axis passing through its center. The angular velocity, , of the sphere in
by the following equations, where is the time in seconds.
What is the angular displacement of the sphere at time
(A)
radian
(B)
radians
(C)
radians
(D)
radians
is given
seconds?
Answer D
Correct. The angular displacement is found by integrating the angular velocity. For the time interval
, the displacement is simply
integrate
or
to get
, which evaluates to
displacements for the two intervals gives
2.
. For the time interval
,
. Adding the
radians.
A disk of radius rotates about an axis passing through its center and perpendicular to the plane of the disk. The
is positive, and the disk is given a constant, negative angular acceleration
. Which of
initial angular speed
the following expressions correctly represents the magnitude of the net linear acceleration for a point located at the
edge of the disk?
(A)
(B)
(C)
(D)
Answer D
Correct. First, the centripetal acceleration can be found by combining the expressions
to get
the angular acceleration by
and
. Next, the tangential acceleration has a magnitude related to
. Finally, the total linear acceleration is found by using the
AP Physics C: Mechanics
Page 1 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
Pythagorean theorem with the centripetal and tangential acceleration components:
.
3.
Three disks are attached together and can rotate together about an axis through their common center, as shown in
meter,
meter, and
meters, and each disk has a force of newtons
the figure. The disks have radii of
exerted at the locations and in the directions shown. What is the magnitude of the net torque about the center of the
three-disk system?
(A)
(B)
(C)
(D)
Answer B
Correct. The inner and outer disks have counterclockwise torques exerted on them, while the middle disk
has a clockwise torque that must be subtracted to get the net torque. Taking the counterclockwise
direction as positive, the torques exerted on the disks are
for the small disk,
for the middle disk, and
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AP Physics C: Mechanics
for the large disk.
Scoring Guide
Unit 5 Progress Check: MCQ
The net torque is the sum of these, or
.
4.
The uniform rod shown in the figure has length and mass . The rod’s rotational inertia is
about its
about Point , which is located a distance from Point . What is the distance ?
center at Point and
(A)
(B)
(C)
(D)
Answer B
Correct. Using the parallel-axis theorem,
.
AP Physics C: Mechanics
Page 3 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
5.
A uniform rod of mass
and length
is supported on a vertical wall at its left end by a pivot and at its center by
is placed on the rod a distance
from the rod’s right
a string, as shown in the figure. A small block of mass
end. What is the magnitude of the force that the pivot exerts on the rod?
(A)
(B)
(C)
(D)
Answer C
Correct. Both the string tension and the force from the pivot are unknown. The vertical or y-component
of the tension is found from applying Newton’s first law in rotational form about the pivot:
.
Because the string is at a 45-degree angle, the horizontal or x-component of tension is the same:
. The y-component of the force from the pivot is found from applying
Newton’s second law in the y-direction:
.
The x-component of the force from the pivot is found from applying Newton’s first law in the x-
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AP Physics C: Mechanics
Scoring Guide
Unit 5 Progress Check: MCQ
direction:
.
The magnitude of the force from the pivot is found by vector addition of the two components via the
Pythagorean theorem:
.
6.
A pulley with a rotational inertia of
and radius
can rotate with negligible friction in its axle. A
block is attached to a lightweight cord that passes over the pulley, while the other end of the cord is pulled with a
constant force of magnitude , as shown in the figure. The string does not slip relative to the pulley. The block
on the cord as it accelerates downward, and the pulley has a clockwise angular acceleration
exerts a force of
. What is
?
(A)
(B)
(C)
(D)
Answer B
Correct. The net torque
exerted on the pulley must be
. This means that the tensions in the vertical and horizontal
sections of the cord differ by an amount
, which is
AP Physics C: Mechanics
Page 5 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
. Since the horizontal cord section has
tension in the vertical section,
is
a tension that is
less than the
.
7.
Points and are located on a disk that rotates about its center, as shown in the figure. The disk has an angular
velocity and an angular acceleration that are both in the counterclockwise direction. How do the magnitudes of the
centripetal and tangential accelerations of the two points compare to each other?
Tangential acceleration
Centripetal acceleration
A
Greater for Point
Greater for Point
B
Greater for Point
Greater for Point
C
Greater for Point
Greater for Point
D
Greater for Point
Greater for Point
(A) A
(B) B
(C) C
(D) D
Answer D
Correct. For a point located a distance from the axis of rotation, the tangential acceleration
and
centripetal acceleration
are both proportional to and therefore larger for Point . Note that the
angular velocity and angular acceleration are the same for all points on the disk.
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AP Physics C: Mechanics
Scoring Guide
Unit 5 Progress Check: MCQ
8.
Four forces with the same magnitude are exerted on a uniform, regular hexagon as shown in the figure. Which of
the labeled forces exerts the greatest torque about the center of the hexagon?
(A)
(B)
(C)
(D)
Answer A
Correct. This force acts with the longest lever arm, since the angle between the force and the position
degrees.
vector (from center to force location) is
AP Physics C: Mechanics
Page 7 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
9.
Two identical cylinders of mass
are attached to a rotating apparatus at equal distances from the rotation axis,
as shown in the figure. The rotational inertia of the apparatus-cylinders system is experimentally determined for
multiple values of . The solid line in the graph shows the best-fit line for as a function of for the identical
cylinders with mass . The experiment is then repeated for two other identical cylinders of unknown mass. The
dashed line in the graph shows the best-fit line for as a function of for the cylinders of unknown mass. Based
?
on the graph, what is the unknown mass in terms of
(A)
(B)
(C)
(D)
Answer C
Correct. For cylinders of mass
the rotational inertia of the system is
, and the
cylinder mass is proportional to the slope of the versus graph. The slope of the graph for the
unknown mass is twice that of the graph for mass , so the unknown mass is
.
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AP Physics C: Mechanics
Scoring Guide
Unit 5 Progress Check: MCQ
10.
A rigid rod of negligible mass is free to rotate on a pivot, as shown in the figures. In experimental Trial a uniform
sphere of mass
is attached to the rod with its center a distance
to the left of the pivot, and a second
is attached to the rod with its center a distance
to the right of the pivot. The radii
uniform sphere of mass
of the spheres are negligible compared to
. The rod is held horizontally and then released. In Trial the spheres
are switched as shown, and the rod is again held horizontally and released. Compare the rotational motion of the
rod-spheres system in the two trials.
Trial 1
Trial 2
A
Remains at rest
Remains at rest
B
Remains at rest
Begins to rotate
C
Begins to rotate
Remains at rest
D
Begins to rotate
Begins to rotate
(A)
(B)
(C)
(D)
AP Physics C: Mechanics
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Scoring Guide
Unit 5 Progress Check: MCQ
Answer B
Correct. In Trial the net torque is zero, since each sphere’s weight results in the same torque of
exerted in opposite directions. In Trial the net torque is not zero (and is in fact clockwise),
since the larger sphere results in a torque of
, which is greater than the
torque of
from the smaller sphere.
11.
A force with magnitude
is applied to the right end of a uniform rod of mass
and length that lies on top of a
horizontal surface, as shown in Figure . The rod is free to rotate with negligible friction about an axis through its
, and
center and perpendicular to the plane of the surface. The rotational inertia of the rod about the axis is
the rod’s initial angular acceleration is . In a second scenario, the same force is applied to the same rod at the
same location, but the axis of rotation is now a distance
from the left end of the rod, as shown in Figure . This
?
time the rod’s initial angular acceleration is . What is the ratio
(A)
(B)
(C)
(D)
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AP Physics C: Mechanics
Scoring Guide
Unit 5 Progress Check: MCQ
Answer B
Correct. The angular acceleration is given by the rotational analog of Newton’s second law,
torque
changes from
factor
in Figure
in Figure
to
. The
in Figure . This is an increase by a
. Using the parallel-axis theorem, the rotational inertia is found to change from
to
, in Figure . This is an increase by a factor of
. Since the torque and rotational inertia change by the same factor, the angular
acceleration
does not change.
12.
A disk of radius moves with negligible friction along a horizontal surface with a translational velocity as it
rotates with an angular velocity . Which of the following statements must be true regarding the angular velocity of
the disk?
(A)
(B)
(C)
(D) A relationship cannot be determined for
in terms of
and
.
Answer D
Correct. Since friction is negligible, the disk can slip on the surface as it moves and rotates. The angular
AP Physics C: Mechanics
Page 11 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
and translational velocities of the wheel have no clear relationship without additional information.
13.
Three solid objects of equal mass, , , and , each hang from a pivot at the top end of the object, as shown in the
figure. Object has its center of mass (
) located midway between the top and bottom ends. Object has its
located closer to the bottom end. Object is identical to Object , except that it is hung upside-down and
is closer to the top end. Which object(s) has (have) the greatest rotational inertia about the pivot?
so its
(A) Object
only
(B) Object
only
(C) Object
only
(D) Objects
and
Answer B
Correct. The mass of Object
is distributed farther from the pivot point than it is for the other two
objects.
, the disk starts to spin about an axis through its center. The disk spins with a
A disk is initially at rest. At time
constant angular acceleration, and after second the disk has an angular displacement of radian.
14.
The disk continues spinning with the same angular acceleration. At what time does the disk have an angular
displacement of radians?
(A)
seconds
(B)
seconds
(C)
seconds
(D)
seconds
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AP Physics C: Mechanics
Scoring Guide
Unit 5 Progress Check: MCQ
Answer A
. In order for the angle to increase
Correct. This result can be determined from the equation
by a factor of , the time must increase by a factor of .
15.
Now the disk is given an initial angular velocity
at time
angular acceleration
. The disk rotates through an angle
the following is true for a disk with an initial angular velocity
(A) The disk stops in a time greater than
(B) The disk stops in a time less than
. This time, the disk decelerates with a constant
in a time
before coming to a stop. Which of
and constant angular acceleration
?
.
.
(C) The disk rotates through an angle greater than
(D) The disk rotates through an angle less than
before stopping.
before stopping.
Answer C
Correct. This result can be determined from the equation
, which for the
original disk becomes
.
and
results in a larger angular displacement. Alternatively,
From this we see that doubling both
both disks must come to a stop in the same time
, with the second disk spinning faster
. Since the second disk spins faster during the same amount of time, it rotates
for all times
through a greater angle.
AP Physics C: Mechanics
Page 13 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
16.
A bar of length and negligible mass is free to rotate about a pivot located a distance
from its right end, as
shown in the figure. Two applied forces,
and
, are exerted on the bar at the locations and in the
directions shown. Which force results in a torque of greater magnitude about the pivot, and what reasoning supports
this claim?
(A)
, because the lever arm is zero for
.
(B)
, because the product
(C)
, because
is greater than
(D)
, because
is directed along the length of the rod.
is greater than
.
.
Answer A
Correct. The line of action for
both zero.
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AP Physics C: Mechanics
goes through the pivot point, so the lever arm and torque for
are
Scoring Guide
Unit 5 Progress Check: MCQ
17.
A beam of uniform mass density can rotate freely about a support pivot located under its midpoint, as shown in the
figure. Three non-zero forces, , , and , are exerted downward on the beam. Forces
and
are applied at
is applied somewhere between the pivot and . When all forces are applied
either end of the beam, and force
the beam stays in rotational equilibrium. Which of the following claims about the force magnitudes must be correct,
and what reasoning supports the claim?
(A)
(B)
, because
has a longer lever arm than
torque of equal magnitude.
, because
, so it must be smaller in order to contribute a
contributes a torque to counteract the torques from both
and the weight of the
beam.
(C)
, because the net force on each side of the pivot must be equal for the beam to be in rotational
equilibrium.
(D)
, because
and
same direction as .
are the same distance from the pivot, and
contributes a torque in the
Answer D
and
contributing torques in the same
Correct. For the beam to be in rotational equilibrium with
direction, the torques from
and
must each be less than that from . Since
and
are
equidistant from the pivot,
must be less than .
AP Physics C: Mechanics
Page 15 of 16
Scoring Guide
Unit 5 Progress Check: MCQ
18.
Two pulleys of different radii are rigidly attached together at their centers, as shown in the figure. The smaller
pulley, of radius
, has a lightweight string wrapped around it. A force of
pulls downward on the left
pulls downward on the right section of string. The larger pulley, of radius
section of string, and a force of
, also has a lightweight string wrapped around it. A single force of
pulls downward on this string. Which
of the following gives the correct direction of the two-pulley system’s angular acceleration, and provides
appropriate supporting evidence?
(A) Clockwise. The sum of the torques from the two
(B)
Clockwise. The sum of the lever arms for the two
force.
(C)
Counterclockwise. The torque from the
forces.
(D) Counterclockwise. The
forces is greater than torque from the
force.
forces is greater than the lever arm for the
force is greater than each of the torques from the
force is greater than the sum of the two
forces.
Answer A
Correct. The angular acceleration, and therefore the direction of rotation after being at rest, is in the same
direction as the net torque. Net torque is found by adding the torques in one direction, and subtracting the
torques in the opposite direction. Taking the clockwise direction as positive, the two 4-newton forces
. The -newton force provides a torque
give a torque of
of
. The net torque is then
, which is in the clockwise
direction.
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AP Physics C: Mechanics