AP® Physics 1
Myers Park High School
Problem Set: Rotational Mechanics
A. Angular Quantities
1. A bike wheel rotates 4.50 revolutions. How many radians has it rotated?
2. A carousel is initially at rest. At t = 0 it is given a constant angular acceleration α =
0.060 rad/s2, which increases its angular velocity for 8 s. At t = 8 s:
a) What is the angular velocity of the carousel?
b) What is the linear velocity of a child riding the carousel who is located 2.5 m from the
center of the ride?
c) What is the linear (tangential) acceleration of the child?
d) What is the centripetal acceleration of the child?
e) What is the magnitude of the total acceleration of the child?
3. A grinding wheel 0.35 m in diameter rotates at a constant 2500 rpm (revolutions per
minute).
a) Calculate is angular velocity in rad/s.
b) What is the linear speed of a point on the outside edge of the grinding wheel?
c) What is the acceleration of a point on the outside edge of the grinding wheel?
4. A bicycle tire with a diameter of 80 cm starts from rest and makes 10 revolutions in 5 s.
a) What is the angular acceleration of the bicycle tire?
b) What is the translational acceleration of a point on the outside edge of the tire?
c) What is the translational acceleration of a point on halfway to the outside edge of the
tire?
1) 28.3 rad
2) a. 0.48 rad/s
b. 1.2 m/s
c. 0.15 m/s2
d. 0.58 m/s
e. 0.60 m/s2
3) a. 2.6 x 102 rad/s
b. 46 m/s
c. 1.2 x 104 m/s2
4) a. 5.024 rad/s2
b. 2.010 m/s2
c. 1.004 m/s2
B. Angular Motion
1. What is the average angular velocity of (a) the second hand, (b) the minute hand, and
(c) the hour hand of a watch? (Hint: Use one revolution as a distance.)
2. A disk, initially rotating at 120 rad/s, is slowed at a constant angular acceleration of
4.00 rad/s2.
a) How much time elapses before the disk comes to rest?
b) Through what angle does the disk rotate before coming to rest? (Express your
answer in radians, revolutions, and degrees)
3. Starting from rest, a solid disk rotates about its axis with a constant angular
acceleration. After 5 s it has rotated through 25 radians.
a) What was the angular acceleration during this time?
b) What was the average angular velocity?
c) What is the angular velocity of the disk at the end of the 5s?
d) Assuming the acceleration does not change, through what additional angle will
the disk turn during the next 5s?
AP® Physics 1
Myers Park High School
Problem Set: Rotational Mechanics
4. Pilots can be tested for the stresses of flying high speed jets in a whirling “human
centrifuge”, which starts from rests and takes 1 minute to make 20 complete
revolutions before reaching its final speed.
a) What is its angular acceleration in rev/min2?
b) What is its final angular speed in rev/min? rev/s? rad/s?
5. A cooling fan is turned off when running at 850 rev/min. It turns 1500 revolutions
before it comes to a stop.
a) What is the fan’s angular acceleration?
b) How long did it take the fan to come to a stop?
6. A ball on the end of a string swings in a horizontal circle once every second. State
whether the magnitude of each of the quantities is zero, constant (but not zero), or
changing?
a) Speed
b) Velocity
c) Angular velocity
d) Centripetal acceleration
e) Angular acceleration
f) Tangential acceleration
1. a) 0.105 rad/s
b) 1.75 x 10-3 rad/s
c) 1.45 x 10-4 rad/s
2. a) 30 s
b) 1800 rad, 287 rev,
1.03 x 105o
4. a) 40 rev/min2
b) 40 rev/min
0.667 rev/s
4.187 rad/s
5. a) -0.42 rad/s2
b) 210 s
3. a) 2 rad/s2
b) 5 rad/s
c) 10 rad/s
d) 75 rad
6. a) constant b) changing
c) constant d) constant
e) zero f) zero
C. Torque
1. A bucket filled with water has a mass of 54 kg and is attached to a rope that is wound
around a 5.0 cm radius cylinder. What magnitude of torque does the bucket produce
around the center of the cylinder?
2. Determine the net torque about the axle of the wheel shown below.
15.0 N
20. cm
10.0 N
32 cm
AP® Physics 1
Myers Park High School
Problem Set: Rotational Mechanics
3. The arm of a crane is 15.0 m long and makes an angle of 20.0 with the horizontal.
Assume that the maximum load for the crane is limited by the amount of torque the
load produces around the base of the arm.
a) What is the magnitude of the maximum torque the carne can withstand if the
maximum load is 450 N?
b) What is the magnitude of the maximum load for this crane at an angle of 40.0?
4. Find the torque exerted on the fishing pole by a fish pulling with a force of 100 N as
shown below.
1) 26.5 mN
2) -2.80 mN
3) a. 6340 mN
b. 552 N
4) 68.4 mN
D. Rotational Equilibrium
1. Two children sit on a uniform seesaw such that a 400. N child is 2.0 m from the
support. How far from the center should the second child of weight 475 N sit in order
to balance the system if the support is at the center of the plank?
2. The 400. N child in problem 6 decides that she would like to seesaw alone. To do so,
she moves the board such that its weight is no longer directly over the fulcrum. She
finds that she will be balanced when she is 1.5 m to the left of the fulcrum and the
center of the plank is 0.5 m to the right of the fulcrum. What is the mass of the
plank?
3. A window washer is standing on a scaffold supported by vertical ropes at each end.
The scaffold weighs 200. N and is 3.0 m long. What is the tension in each rope when
a 71 kg worker stands 1.0 m from one end?
4. A 0.100 kg meter stick is supported at its 40.0 cm mark by a string attached to the
ceiling. A 0.700 kg mass hangs vertically from the 5.00 cm mark. A mass is attached
somewhere on the meter stick to keep it horizontal and in both rotational and
AP® Physics 1
Myers Park High School
Problem Set: Rotational Mechanics
translational equilibrium. If the force applied by the string attaching the meter stick
to the ceiling is 19.6 N, what is the value of the unknown mass and where is it
attached to the meter stick?
5. A uniform plank of length 2.0 m and mass 30.0 kg is supported by three ropes as
shown below. Find the tension in each rope when the 700 N person is 0.50 m from
the left end.
1) 1.68 m
2) 122 kg
4) Mass 1.2 kg
Location 59.58 cm
5) T1 = 501 N
T2 = 672 N
T3 = 384 N
3) 564 N; 332 N
E. Rotational Dynamics
1. A wrench 0.25 m long is used to loosen a bolt. A force of 400N is applied to the
wrench.
a) What torque is applied to the bolt?
b) What force is needed to create the same torque using a wrench 0.15m long?
2. A 2 kg solid disk with a radius of 0.22 m has a tangential force of 300N applied to it.
a) What is the torque acting on the disk?
1
b) What is the moment of inertia of the disk? ( 𝐼 = 2 𝑀𝑅 2 )
c) What angular acceleration is produced by the torque?
d) If the disk starts from rest and the acceleration is constant for 3.0s, what is the
angular velocity of the disk at the end of 3.0s?
e) Through what angle in radians has the disk rotated during this time?
AP® Physics 1
Myers Park High School
Problem Set: Rotational Mechanics
Ffr
33 cm
FT
3. A 15.0 N force FT is applied to a cord wrapped around a pulley of mass M and radius
R. The pulley accelerates from rest to an angular speed of 30 rad/s in 3.00 s. If there
is a frictional torque τfr = 1.10 m*N at the axle, use Newton’s 2nd law for rotational
motion to determine the moment of inertia of the pulley. Assume the pulley rotates
about its center.
4. Consider the pulley from the previous problem except this time instead of a constant
15.0 N force, we have a bucket that weighs 15.0 N. If there is a frictional torque τfr =
1.10 m*N at the axle, determine the angular acceleration α of the pulley and the linear
acceleration a of the bucket.
2.
1. a) 100 mN
b) 667 N
a) 66 mN
b) 4.84 x 10-2 kg*m/s2
c) 1.36 x 103 rad/s2
d) 4.08 x 103 rad/s
e) 6.12 x 103 rad
3. 0.385 kg*m2
4. α = 6.98 rad/s2
a = 2.30 m/s2
F. Rotational Kinetic Energy
1. A centrifuge rotor has a moment of inertia of 3.75 x 10-2 kg*m2. What is the
change in rotational kinetic energy as it starts from rest and reaches 8250
revolutions per minute?
2. A bowling ball mass of 7.3 kg and a radius of 9.0 cm rolls without slipping down
2
a lane at 3.3 m/s. What is its total kinetic energy? (𝐼 = 5 𝑀𝑅 2 )
DIAGRAM IS FOR QUESTION #3.
H
2
3. Derive an equation to define the speed of a solid sphere (𝐼 = 5 𝑀𝑅 2 ) of mass M
and radius R when it reaches the bottom of an incline if it starts from rest at a
vertical height H and rolls without slipping. See diagram above.
1. 1.4 x 104 J
2.
56 J
10
3. 𝑣 = √ 7 𝑔𝐻
AP® Physics 1
Myers Park High School
Problem Set: Rotational Mechanics
G. Angular Momentum
1. a) What is the angular momentum of a 2.8 kg uniform grinding wheel disk of
1
radius 18 cm ( 𝐼 = 2 𝑀𝑅 2 ) when rotating at 1500 rpm (revolutions per minute)?
b) How much torque is required to stop it in 6.0 s?
2. A diver can reduce their moment of inertia by a factor of about 3.5 when changing
from the straight position to the tuck position. If the diver makes 2.0 revolutions
in 1.5 s while in the tuck position, what is their angular speed (in rev/s) when in
the straight position?
3. A nonrotating cylindrical disk of moment of inertia I is dropped on an identical
rotating disk at an angular speed ω. Assuming no external torques, what is the
final common speed of the two disks?
4. A person of mass 75 kg stands at the center of a rotating merry-go-round platform
radius 3.0 m and a moment of inertia of 920 kg*m2. The platform rotates without
friction with an angular velocity of 2.0 rad/s. The person walks radially (along the
radius) to the edge of the platform.
a) Calculate the angular velocity when the person reaches the edge.
b) Calculate the rotational kinetic energy of the system person plus the
platform before and after the walk.
5. Astronomers often detect stars that are rotating extremely rapidly, known as
neutron stars. These stars are believed to have formed in the inner core of a larger
star that collapsed, due to its own gravitation, to a star of a very small radius and
very high density. Before collapse suppose the core of such star is the size of our
Sun (R = 7 x 105 km) with mass 2.0 times as great as the Sun, and is rotating at a
speed of 1 revolution every 10 days. If it were to undergo gravitational collapse to
a neutron star of radius 10 km, what would its rotational speed be? Assume the
2
star is a uniform solid sphere at all times. ( 𝐼 = 5 𝑀𝑅 2 )
1. 7.122 kg*m2/kg
4. a. 1.2 ras/s
b. 1.8 x 103 J;
1.1 x 103 J
2. 0.381 rev/s
5. 6 x 103 rev/s
3. ω/2