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ROTATIONAL MOTION extra problems (honors)
1)
A major-league pitcher throws a baseball toward home plate. The ball rotates at 1560 rpm, and it travels the
18.5 meters to the plate at an average translational speed of 40.2 m/s. How many revolutions does the ball
make during this trip?
2)
A sphere of radius 60.0 cm rotates about a fixed axis through its center. A point on the surface of the sphere
30.0 cm from the axis moves a distance of 12.0 cm in a time interval of 7.00 s. Through what angular
distance has the sphere rotated during this time interval?
3)
A disk initially rotating at 125 rad/s is slowed down with a constant angular acceleration of magnitude
4.00 rad/s².
a. What is the original linear speed of a point on the disk that is 0.450 m from the axis?
b. How much time elapses before the disk stops?
c.
4)
Through what angle did the disk rotate during this time?
A 1.20-m diameter flywheel is rotating clockwise at 215 rpm.
a. What is the linear speed of a point on the rim of the flywheel?
b. What constant angular acceleration will cause the wheel to increase its speed to 915 rpm in 60.0 s?
c.
5)
Through what linear distance does the outer point of the wheel travel during the 60.0-s interval?
After exiting a helicopter, you notice that the rotor's motion changed from 300. rpm to 225 rpm in
1.00 minutes. Each blade of the rotor is 2.95 meters in length.
a. Find the magnitude of the angular acceleration of the rotor.
b. Find the magnitude of the linear acceleration of the tip of the rotor.
c.
Calculate the angular distance through which the rotor moved during the 1.00-minute time interval.
6)
A wheel of radius 25.0 cm initially rotates clockwise about a fixed axis through its center with an initial speed
of 1.20 rad/s. (a) What is the linear speed of a point on the end of the wheel? (b) If it accelerates uniformly to
2.00 rad/s counterclockwise during a 5.00 s interval, what is the angular displacement of the wheel during this
interval?
7)
When a torque of magnitude 32.0 N·m is applied to a certain wheel, it acquires an angular acceleration of
25.0 rad/s². What is the moment of inertia of the wheel?
8)
A flywheel (disk) initially spinning at 1850 rpm counterclockwise is brought to rest in 3.00 minutes. (a) What is
the wheel’s average angular acceleration? (b) What is the angular displacement of the flywheel during this
acceleration? (c) If the flywheel has a mass of 155 kg and a radius of 0.650 m, what torque was required to
accelerate it?
9)
A heavy flywheel rotating counterclockwise about its middle at an original speed of 157.1 rad/s is slowing
down because of friction in its bearings.
a. If its acceleration is constant at 1.785 rad/s², find its angular velocity at the end of 1.00 minute.
b. The flywheel is 5.50 m in diameter and has a mass of 985 kg. If the flywheel resembles a uniform disk,
find the torque applied by friction.
c.
The bearings are 0.100 m from the axis. Calculate the amount of frictional force the bearings apply to the
wheel. (assume θ = 90°)
10) A torque of 35.0 Nm, out, is applied to a door of length 1.05 m and mass 45.0 kg to open it. (a) Calculate the
angular acceleration of the door. (b) Calculate the magnitude of the linear acceleration of a point on the
outside edge of the door. (c) Calculate the angular velocity of the door when it has rotated through an angle
of 0.555 rad. (d) Calculate the linear speed of a point on the outside edge of the door at that instant.
11) A uniform solid sphere has a mass of 1.765 kg and a radius of 0.854 m.
a. Find the torque required to bring the sphere from rest to an angular velocity of 317 rad/s, clockwise, in
15.5 s.
b. What magnitude force applied tangentially at the equator would provide the needed torque?
12) Calculate the magnitude of the angular momentum of the earth due to its daily rotation about its own axis.
Assume the earth is a uniform solid sphere.
(massearth = 5.98 x 1024 kg , equatorial radius of Earth = 6.378 x 103 km)
13) Calculate the amount of angular momentum the earth has due to its revolution about the sun. Assume its
orbit is circular.
(massearth = 5.98 x 1024 kg , average distance of Earth from Sun = 1.5 x 108 km)
14) A playground merry-go-round makes 4.00 rotations in 6.00 s. (a) Calculate its angular speed. (b) If the
merry-go-round is disk-shaped, with a mass of 115 kg and a diameter of 3.50 m, calculate its angular
momentum. (It rotates counterclockwise as viewed from above.) (c) Calculate its rotational kinetic energy.
(d) If some children jump onto the ride, causing its moment of inertia to increase by 25.0%, what is its new
angular speed?
15) Suppose that the sun runs out of nuclear fuel and suddenly collapses to form a so-called white dwarf star, with
a diameter about equal to that of the earth. Assuming no mass loss, what would then be the new rotation
period of the sun, which is currently about 25 days? (masssun = 2.0 x 1030 kg , equatorial radius of sun =
7.0 x 105 km , equatorial radius of Earth = 6.378 x 103 km)
16)
A piece of machinery with a moment of inertia of 2.5 kg∙m² rotates with a constant angular speed of 4.0 rad/s.
A bolt loosens, causing a part of it to shift outward from the rotation axis, increasing the moment of inertia by
1.0 kg∙m². What would the new angular speed of the machinery be?
17) A man stands on a rotating platform that has an angular speed of 6.28 rad/s; his arms are outstretched and he
holds a weight in each hand. With his hands in this position the total moment of inertia of the man, the
weights, and the platform is 6.00 kg·m². If by moving the weights the man decreases the total moment of
inertia to 2.00 kg·m²,
a. What is the resulting angular speed of the platform?
b. By how much is the kinetic energy increased?
18) Calculate the moment of inertia of a wheel that has a rotational kinetic energy of 24,400 J when it is rotating at
62.8 rad/s.
19) The oxygen molecule, O2, has a total mass of 5.30 x 10-26 kg and a moment of inertia of 1.94 x 10-46 kg·m².
Suppose that such a molecule in a gas has a translational speed of 500.0 m/s and that its rotational kinetic
energy is 2/3 of its translational kinetic energy. Find its angular speed.
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20) A solid 0.75-kg ball rolls without slipping – which means that the average linear speed v  r of a point on
the edge of the ball is equal to the linear speed of the ball’s center of mass – on a level surface. If the center
of mass of the ball moves in a straight line at a speed of 0.50 m/s, what is its total kinetic energy?
21) A 1.5-hp motor drives a machine disk having a mass of 4.5 kg and a diameter of 0.40 m from rest to a speed
of 750 rpm. How long does it take to reach the operating speed?
22) A uniform, solid sphere of mass 350 grams and radius 25.0 cm has an axle attached to it tangent to its
surface. The axle is oriented horizontally, causing the sphere to be suspended below it, its center of mass
directly below the axle. (See diagram below left.) Someone lifts the sphere until its center of mass is at the
same vertical level as the axle. Upon releasing it, the sphere begins to rotate counterclockwise about the
axle. (See diagram below right.) Calculate the linear speed of a point on its outer edge (farthest from the axle)
when the sphere’s center of mass returns to its position directly below the axle.
23) A hoop starts from rest at a height 1.2 m above the base of an inclined plane and rolls down (without slipping)
to the bottom. What is the linear speed of the hoop’s center of mass just as the hoop leaves the bottom of the
incline?
24) A hollow sphere with a radius of 15 cm rolls without slipping on a level surface with a constant angular speed
of 10. rad/s. To what vertical height on an inclined plane will the sphere roll before coming to rest? (neglect
frictional losses)
25) A solid steel ball of radius r rolls without slipping down an incline into a vertical loop of radius R. (just like the
Hot Wheels setup used during the unit on Energy) In order to maintain contact with the track while at the top
v  gR
of the loop, the ball’s center of gravity must be moving at a minimum linear speed of
. Find the
height the ball must start relative to the bottom of the loop in order to stay on the track in terms of the other
variables identified.
ANSWERS:
1. 12.0 rev
2. 0.400 rad
3. a. 56.3 m/s b. 31.3 s c. 1960 rad
4. a. 13.5 m/s b. 1.22 rad/s², in c. 2130 m
5. a. 0.131 rad/s² b. 0.386 m/s² c. 1650 rad
6. a. 0.300 m/s b. 2.00 rad, out
7. 1.28 kg·m²
8. a. 1.08 rad/s², in b. 17,400 rad, out c. 35.4 Nm,
in
9. a. 50.0 rad/s, out b. 6650 N·m, in c. 66500 N
10. a. 2.12 rad/s², out b. 2.23 m/s² c. 1.53 rad/s,
out
d. 1.61 m/s
11. a. 10.5 N·m, in b. 12.3 N
12. 7.10 x 1033 kg·m²/s
13. 2.68 x 1040 kg·m²/s
14. a. 4.19 rad/s
b. 738 kgm²/s, out
3.35 rad/s
15. 180 seconds
16. 2.9 rad/s
17. a. 18.8 rad/s b. 235 J
18. 12.4 kg·m²
19. 6.75 x 1012 rad/s
20. 0.13 J
21. 0.25 s
22. 01.87 m/s
23. 3.4 m/s
24. 0.19 m
25. h = (27/10) R
c. 1550 J
d.