Review Torque
MC Section:
1. A solid uniform sphere, of mass M and radius R is pivoted about an axis that is tangential to its
surface. It’s moment of inertia is:
a. MR2
b. (2/5)MR2
c. (3/5)MR2
d. (5/2)MR2
e. (7/5)MR2
2. A disk with a rotational inertia of 5.0 kg m2 and a radius of 0.25 m rotates on a frictionless fixed
axis perpendicular to the disk and through its center. A force of 8.0 N is applied in the plane of
the disk and tangent to the rim. If the disk starts at rest, then after it has turned through half a
revolution its angular velocity is:
a. 0.57 rad/s
b. 0.64 rad/s
c. 0.80 rad/s
d. 1.6 rad/s
e. 3.2 rad/s
3. A 16 kg block is attached to a cord that is wrapped around the
rim of a flywheel of diameter 0.4 m and hangs vertically as
shown. The rotational inertia of the flywheel is 0.50 kg m2.
Then the block is released and the cord unwinds, the
acceleration of the block is:
a. 0.15g
b. 0.56g
c. 0.84g
d. g
e. 1.3g
For problems 4-6 refer to the picture below of a strut and hanging mass
4. A 5.0 m weightless strut, hinged to the wall, is used to support an 800 N block as shown. What is
the tension force on the 3.0 m string?
a. 600 N
b. 750 N
c. 800 N
d. 1000 N
3m
e. 1200 N
5. If the strut has a weight of 400 N what is the tension on the string?
a. 600 N
b. 750 N
c. 800 N
d. 1000 N
e. 1200 N
800N
6. The horizontal force on the hinge for the strut is closest to:
a. 600 N
b. 750 N
c. 800 N
d. 1000 N
e. 1200 N
7. A wheel rolls without slipping along a horizontal road as shown. The wheel has a center of mass
velocity of 12 m/s and a radius of 0.5 m. The instantaneous velocity of point P is:
ω
a. 6 m/s
b. 12 m/s
c. 24 m/s
vcm
d. zero
e. not enough information
P
8. A man, holding a weight in each hand, stands at the center of a horizontal frictionless rotating
turntable. The effect of the weights is to double the rotational inertia of the system consisting of
the man, the turntable, and the weights. As he is rotating, the man opens his hands and drops the
two weights. They fall outside the turntable. As a result:
a. his angular velocity doubles
b. his angular velocity remains the same
c. his angular velocity is halved
d. the direction of his angular momentum vector changes
e. none of the above occur
F
F
F
9. A uniform disk, a thin hoop and a uniform sphere, all with the same mass and same radius are each
free to rotate about an axis through its center. Assume the hoop is connected to the central axis
by thin spokes of negligible mass. Identical forces are simultaneously applied to the rims as
shown. All objects start from rest. Rank the objects according to their angular momenta after a
given time t, least to greatest.
a. All the same
b. Disk, hoop, sphere
c. Hoop, disk, sphere
d. Hoop, sphere, disk
e. Hoop, disk, sphere
MC Answers:
1. E 2. D 3. B 4. A 5. B 6. B 7. D 8. B 9. A
FR Section:
1. (1999 APC Mech Test) As shown above, a uniform disk is mounted to an axle and is free to
rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate
with the disk. A block is attached to the end of the rod. Properties of the disk, rod, and block
are as follows:
Disk:
Rod:
Block:
mass = 3m, radius = R, rotational inertia about center ID = 3/2mR2
mass = m, length = 2R, rotational inertia about one end is IR = 4/3mR2
mass = 2m
The system is held in equilibrium with the rod at an angle θ0 to the vertical, as shown above,
by a horizontal string of negligible mass with one end attached to the disk and the other to a
wall. Express your answers to the following in terms of m, R, θ0, and g
(a) Determine the tension in the string.
The string is now cut, and the disk-rod-block system is free to rotate.
(b) Determine the following for the instant immediately after the string is cut.
i. The magnitude of the angular acceleration of the disk.
ii. The magnitude of the linear acceleration of the mass at the end of the rod.
As the disk rotates, the rod passes the horizontal position shown above.
(c) Determine the linear speed of the mass at the end of the rod for the instant the rod is in
the horizontal position.
2. A system is composed of two identical disks. Initially, the bottom disk is rotating with angular
velocity ω0 around a fixed frictionless axle, and the top disk is at rest. The top disk is now
dropped onto the bottom one, and eventually both disks rotate with the same final angular
velocity ωf (around the same axis). There is no net external torque acting on the two-disk system.
a. Write an expression for ωf in terms of ω0.
b. Has the rotational kinetic energy of the system changed? If so, how?
3. A bowling ball is thrown with an initial speed of 9m/s. Initially it slides along the lane
without spinning, but eventually catches and starts spinning. The coefficient of kinetic
friction between the ball and the lane is 0.24.
a. What is the balls linear acceleration and angular acceleration?
vcm
t=0
ω
b. How long until the ball starts rolling without slipping?
t=?
c. How far has the ball slid?
d. What is its linear speed and angular speed when the ball begins smooth rolling
(i.e. rolling without slipping)