AP Physics Rotational Motion – Part I

Introduction: Which moves with greater speed on a merry-go-round - a horse near the center or one near the outside? Your

answer probably depends on whether you are considering the translational or rotational motion of the horses. Have you ever

linked arms with friends at a skating rink while making a turn? If you have, you probably noticed that the person on the

inside moved very little while the person on the outside had to run to keep up. The outside person traveled a greater distance

per period of time and therefore had the greater translational speed. During the same period of time all skaters rotated

through the same angle per period of time and had the same rotational speed.

In our previous study of motion we discussed translational motion - that is the motion of bodies moving as a whole

without regard to rotation. In this unit we will extend our ideas of motion to include the rotation of a rigid body about a fixed

axis. If the axis is inside the body we tend to say the body rotates about its axis. If the axis is outside the body, we say the

body revolves about an axis. An example of this would be the earth which daily rotates about its axis and yearly revolves

around the sun.

An object rotating about an axis tends to remain rotating about the same axis unless acted upon by a net external

influence. This property of a body to resist changes in its rotational state is called rotational inertia. The rotational inertia of

a body depends on the amount of mass the body possesses and on the distribution of that mass with respect to the axis of

rotation. The greater the distance of the bulk of the mass from the axis of rotation - the greater the rotational inertia.

A long pendulum has a greater rotational inertia than a short one. The period of a pendulum is directly proportional

to the square root of the length of the pendulum. It takes more time to change the rotational inertia of a long pendulum as it

swings back and forth. People and animals with long legs tend to walk with slower strides than those with short legs for the

same reason. Have you ever tried running with your legs straight?

Performance Objectives: Upon completion of the readings and activities of the unit and when asked to respond either orally

or on a written test, you will be able to:

state relationships between linear and angular variables.

recognize that the rotational kinematics formulas are analogous to the translational ones. Use these formulas to

solve problems involving rotating bodies.

define rotational inertia or moment of inertia. Calculate the rotational inertia for a point mass, a system of point

masses, and rigid bodies.

use the parallel axis theorem to find the moment of inertia about an axis other than the center of mass.

calculate the kinetic energy of a rotating body.

define torque. Calculate the net torque acting on a body.

state Newton’s second law for rotation. Recognize that the rotational dynamics formulas are analogous to the

translational ones. Use these formulas to solve problems involving rotating bodies.

use the work-kinetic energy theorem for rotation to solve problems.

Textbook Reference: Tipler: Chapter 9

Glencoe Physics: Chapter 8

"To every thing -- turn, turn, turn

there is a season -- turn, turn, turn

and a time for every purpose under heaven."

-- The Byrds (with a little help from Ecclesiastes)

Recall: From the definition of a radian (arc length/radius) θ = s/r, where s is the arc length, r is the radius and θ is the angle

measure in radians. The following quantities are called the bridges between linear and angular measurements: s = rθ

v = rω aT = rα aR = v2/r = rω2

Definitions and Conversions: 1. What angle in radians is subtended by an arc 3.0 m in

length, on the circumference of a circle whose radius is2.0

m? 1.5 rad

2. What angle in radians is subtended by an arc of length

78.54 cm on the circumference of a circle of diameter

100.0 cm? What is the angle in degrees?

1.57 rad 90˚

3. The angle between two radii of a circle of radius 2.00

m is 0.60 rad. What length of arc is intercepted on the

circumference of the circle by the two radii? 1.2 m

4. What is the angular velocity in radians per second of a

flywheel spinning at the rate of 7230 revolutions per

minute? 757 rad/sec

5. If a wheel spins with an angular velocity of 625 rad/s,

what is its frequency in revolutions per minute?

5968 rpm

6. Compute the angular velocity in rad/s, of the

crankshaft of an automobile engine that is rotating at 4800

rev/min. 503 rad/sec

Rotational Kinematics: Rotational motion is described

with kinematic formulas just like the translational motion

formulas. To get the rotational kinematic formulas,

substitute the rotational variables.

7. A flywheel accelerates uniformly from rest to an

angular velocity of 94 radians per second in 6.0 seconds.

What is the angular acceleration of the flywheel in radians

per second squared? 16 rad/s2

8. a) Calculate the angular acceleration in radians per

second squared of a wheel that starts from rest and attains

an angular velocity of 545 revolutions per minute in 1.00

minutes. b) What is the angular displacement in radians

of the wheel during the first 0.500 minutes? c) During

the second 0.500 minutes?

0.95 rad/s2 428 rad. 1283 rad

9. Find the angular displacement in radians during the

second 20.0 second interval of a wheel that accelerates

from rest to 725 revolutions per minute in 1.50 minutes?

506 radians

10. A fly wheel requires 3.0 seconds to rotate through

234 rad. Its angular velocity at the end of this time is 108

rad/s. Find a) the angular velocity at the beginning of the

3 second interval; b) the constant angular acceleration.

48 rad/s 20.0 rad/s2

11. A playground merry-go-round is pushed by a child.

The angle the merry-go-round turns through varies with

time according to θ(t) = 2t + 0.05t3, where θ is in radians

and t is in seconds. a) Calculate the angular velocity of the

merry-go-round as a function of time. ω = 2 + 0.15t2

b) What is the initial value of the angular velocity?

2 rad/s

c) Calculate the instantaneous velocity at t = 5.0 sec.

d) Calculate the average angular velocity for the time

interval t = 0 to t = 5 seconds. 5.75 rad/s 3.25 rad/s

12. A rigid object rotates with angular velocity that is

given by ω = 4 + 8t2, where ω is in rad/sec and t is in

seconds. a) Calculate the angular acceleration as a

function of time. b) Calculate the instantaneous angular

acceleration a at t = 2 sec.

c) Calculate the average angular acceleration for the time

interval t = 0 to t = 2seconds.

α = 16t 32 rad/s2 16 rad/s2

13. A bicycle wheel of radius 0.33 m turns with angular

acceleration α = 1.2 – 0.4t, where α is in rad/s2 and t is in

seconds. It is at rest at t = 0.

a) Calculate the angular velocity and angular

displacement as functions of time.

b) Calculate the maximum positive angular velocity and

maximum positive angular displacement of the wheel.

ω = 1.2t - 0.2t2 θ = 0.6t2 -0.067t3 1.8 rad/s 7.2 rad

14. A roller in a printing press turns through an angle θ

given by θ(t) = 2.50t2 – 0.400t3.

a) Calculate the angular velocity of the roller as a function

of time. ω(t) = 5t – 1.2t2

b) Calculate the angular acceleration of the roller as a

function of time. α(t) = 5 – 2.4t

c) What is the maximum positive angular velocity and at

what value of t does it occur?

5.21 rad/s 2.08 sec

15. a) A cylinder 0.15 m in diameter rotates in a lathe at

750 rpm. What is the tangential velocity of a point on the

surface of the cylinder? b) The proper tangential velocity

for machining cast iron is about 0.60 m/s. At how many

rpm should a piece of stock 0.05 m in diameter be rotated

in a lathe? 5.89 m/s 229 rev/min

16. Find the required angular velocity of an

ultracentrifuge in rpm for the radial acceleration of a point

1.00 cm from the axis to equal 300,000g (that is 300,000

times the acceleration of gravity). 1.64 x 105 rev/min

17. A wheel rotates with a constant angular velocity of 10

rad/s. a) Compute the radial acceleration of a point 0.5 m

from the axis using the relation, a⊥ = rω2. 50 m/s2

b) Find the tangential velocity of the point, and compute

its radial acceleration from the relation,

ac = v2/r. 5m/s 50 m/s2

Rotational Inertia is the resistance of a rotating body to

changes in its angular velocity. According to Newton's

First Law a body tends to resist a change in its motion.

The amount of inertia a body possesses is directly related

to the mass. For rotational motion, an analogous

situation exists. However, rotational inertia depends on

the mass and on the distribution of the mass about the

axis of rotation. This quantity that relates mass and

position of the mass relative to the axis of rotation

is called the moment of inertia and has units of kg-m2.

The symbol for moment of inertia is I. For a point mass m

a distance r from the axis of rotation, the moment of

inertia will be I = mr2. For bodies made up of several

small masses just add all the moments of inertia together.

For bodies which are not composed of discrete point

masses but are continuous distributions of matter, the

methods of calculus must be used to find the moment of

inertia. The moments of inertia for a few simple but

important rigid bodies of uniform composition are listed

on Page 295 in Tipler Physics.

18. Small blocks, each of mass 2.0 kg, are clamped at the

ends and at the center of a light rod 1.2 m long. Compute

the moment of inertia of the system about an axis passing

through a point one-third of the length from one end of

the rod if the moment of inertia of the light rod can be

neglected. 1.68 kg-m2

19. Four small spheres, each of mass 0.200 kg, are

arranged in a square 0.400 m on a side and connected by

light rods of negligible mass. Find the moment of inertia

of the system about an axis

a) perpendicular to the plane of the square through the

center. 0.0640 kg-m2

b) bisecting two opposite sides of the square.

0.0320 kg-m2

20. What is the rotational inertia of a solid ball 0.50 min

radius that weighs 80.0 N if it is rotated about a diameter?

0.816 kg-m2

21. What is the rotational inertia of a thick ring that is

rotating about an axis perpendicular to the plane of the

ring passing through its center? The ring has a mass of

1.20 kg and a diameter of 45.0 cm. The hole in the ring is

15.0 cm wide. 0.0340 kg-m2

22. Find the moment of inertia about each of the

following axes for a rod that is 4.00 cm in diameter and

2.00 m long and has a mass of 8.00 kg. a) An axis

perpendicular to the rod and passing through its center.

b) An axis perpendicular to the rod and passing through

one end. c) A longitudinal axis passing through the center

of the rod.

2.67 kg-m2 10.67 kg-m2 0.0016 kg-m2

23. A wagon wheel is constructed as shown in the figure.

The radius of the wheel is

0.300 m and the rim has a mass of

1.20 kg. Each of the eight spokes,

which lie along a diameter and are

0.300 m long has a mass of 0.373 kg.

What is the moment of inertia of the

wheel about an axis through its center and perpendicular

to the plane of the wheel? 0.196 kg-m2

Parallel Axis Theorem: The moment of inertia of any

object about an axis through its center of mass is the

minimum moment of inertia for an axis in that direction of

space. The moment of inertia about any axis parallel to

that axis through the center of mass is given by:

Iparallel axis = Icom + Md2

…where Iparallel axis is the moment of inertia about an new

axis (parallel to the original axis of rotation), Icom is the

moment of inertia about the center of mass, M is the mass

of the object and d is the distance between the original

axis of rotation about the center of mass and the new

proposed axis of rotation.

24. Use the parallel axis theorem to calculate the moment

of inertia of a uniform thin rod of mass M and length l for

an axis perpendicular to the rod at one end. Ml2/3

25. Use the parallel axis theorem to calculate the moment

of inertia of a square sheet of metal of side length a and

mass M for an axis perpendicular to the sheet and passing

through one corner. 2Ma2/3

26. Find the moment of inertia of a thin-walled hollow

cylinder of mass M and radius R about an axis

perpendicular to its plane at an edge of the cylinder.

2MR2

27. A thin, rectangular sheet of steel is 0.30 m by 0.40 m

and has a mass of 16.0 kg. Find the moment of inertia

about an axis

a) in the plane of the sheet, through the center, parallel to

the long sides. 0.12 kg-m2

b) in the plane of the sheet, through the center, parallel to

the short sides. 0.21 kg-m2

c) perpendicular to the sheet and through the center.

0.33 kg-m2

28. A uniform, thin rod is bent into a square of side

length a. If the total mass is M, find the moment of

inertia about an axis through the center, perpendicular to

the plane of the square. Ma2/3

29. Which pendulum has more rotational inertia, a long

or short one?

30. The four objects shown in the figure below have

equal masses m.

Object A is a solid cylinder of radius R. Object B is a

hollow, thin cylinder of radius R. Object C is a solid

square whose length of side = 2R. Object D is the same

size as C, but hollow (i.e., made up of four thin sticks).

The objects have axes of rotation perpendicular to the

page and through the center of gravity of each object.

a) Which object has the smallest moment of inertia?

b) Which object has the largest moment of inertia?

Kinetic Energy of Rotation: Because a rotating rigid

body consists of particles in motion, it has kinetic energy.

This kinetic energy is computed using the moment of

inertia of the body and the angular velocity. KE = ½Iω2

31. The rotor of an electric motor has a rotational inertia

of 45 kg-m2. What is its kinetic energy if it turns at 1500

revolutions per minute? 555 kJ

32. A grinding wheel in the shape of a solid disk is 0.200

m in diameter and has a mass of 3.00 kg. The wheel is

rotating at 3600 rev/min about an axis through its center.

a) What is its kinetic energy? 1066 J

b) How far would it have to drop in free fall to acquire the

same kinetic energy? 36.3 m

33. The flywheel of a gasoline engine is required to give

up 300.0 J of kinetic energy while its angular speed

decreases from 660 rev/min to 540 rev/min. What

moment of inertia is required for the wheel? 0.380 kg-m2

34. A phonograph turntable has a kinetic energy of

0.0700 J when turning at 78 rpm. What is the moment of

inertia of the turntable about the rotation axis?

35. Energy is to be stored in a large flywheel in the shape

of a disk with radius of 1.20 m and a mass of 80.0 kg. To

prevent structural failure of the flywheel, the maximum

allowed radial acceleration of a point on its rim is 5000

m/s2. What is the maximum kinetic energy that can be

stored in the flywheel? 1.20 x 105 J

36. A light flexible rope is wrapped several times around

a 160.0 N solid cylinder with a radius of 0.25 m. The

cylinder which can rotate without friction about a fixed

horizontal axis is initially at rest. The free end of the rope

is pulled with a constant force P for a distance of 5.00 m.

What must P be for the final speed of the end of the rope

to be 4.00 m/s?

37. Two blocks, one of mass 4.0 kg and the other of mass

2.0 kg are connected

by a light rope that passes

over a pulley as shown in the

figure to the right. The pulley

has radius 0.20 m and moment

of inertia 0.32 kg-m2. The

rope does not slip on the

pulley rim. The larger mass

is 5.0 m above the floor and

released from rest. Use

energy methods to calculate the velocity of the 4-kg block

just before it strikes the floor. 3.74 m/s

Torque: To change the translational inertia of a body

you have to apply a net external force. To change the

rotational inertia of a body you have to apply a torque

(rhymes with fork). If you studied torque in previous

science courses it was probably defined as the product of

the force and the length of the torque arm. The torque

arm (sometimes called lever arm) is the perpendicular

distance between the line of action of the force and the

axis of rotation.

In order to solve problems involving torque, you need to

understand how torque is calculated and then be able to

calculate the net torque acting on a body.

38. Calculate the torque (magnitude and direction) about

point 0 due to the force F in each of the situations

sketched in the figure. In each case the object to which

the force is applied has length 4.00 m, and the force

F =20.0 N.

a) 80.0 m-N ccw b) 69.3 m-N ccw

c) 40.0 m-N ccw d) 34.6 m-N cw e) 0 f) 0

39. Calculate the resultant torque about point O for the

two forces applied in the figure below. 28 m-N cw

40. Calculate the net torque (magnitude and direction) on

the beam shown in the figure below about

a) an axis through O, perpendicular to the figure.

29.5 m-N ccw

b) an axis through C, perpendicular to the figure.

35.6 m-N ccw

41. A small ball of mass 0.75 kg is attached to one end of

a 1.25 meter long massless rod, and the other end of the

rod is hung from a pivot. When the resulting pendulum

is30̊ from the vertical, what is the magnitude of the torque

about the pivot? 4.6 m-N

42. Find the net torque on

the wheel in the figure

about the axle through O if

a = 10 cm and b = 25 cm.

3.55 m-N cw

43. Find the mass M needed to balance the 150-kg truck

on the incline shown in the figure below. The angle of

inclination θ is 45º. Assume all pulleys are frictionless

and massless. 17.7 kg

Rotational Dynamics: In studying translational

dynamics we made use of Newton's Second Law, which

related the acceleration of a body and the forces applied

to the body. An analogous relationship exists between

angular acceleration and a quantity we call a torque.

Qualitatively speaking, torque is the tendency of a force

to cause a rotation of the body on which it acts.

Mathematically speaking, torque is defined as the cross

product of the moment arm and the applied force. The

moment arm is the perpendicular distance between the

force applied and the axis of rotation. The unit for torque

is a meter-newton. The symbol for torque is the lower-

case Greek letter tau, τ.

44. A net force of 10.0 N is applied tangentially to the

rim of a wheel having a 0.25 m radius. If the rotational

inertia of the wheel is 0.500 kg m2, what is its angular

acceleration? 5 rad/s2

45. A solid ball is rotated by applying a force of 4.7 N

tangentially to it. The ball has a radius of 14 cm and a

mass of 4.0 kg. What is the angular acceleration of the

ball? 21 rad/s2

46. A fly wheel in the shape of a thin ring has a mass of

30.0 kg and a diameter of 0.96 m. A torque of 13 m-N is

applied tangentially to the wheel. How long will it take

for the flywheel to attain an angular velocity of 10.0

rad/s? 5.3 sec

47. A cord is wrapped around the rim of a flywheel

0.5 m in radius, and a steady pull of 50.0 N is exerted on

the cord. The wheel is mounted with frictionless bearings

on a horizontal shaft through its center. The moment of

inertia of the wheel is 4.0 kg-m2. Compute the angular

acceleration of the wheel. 6.25 rad/s2

48. A grindstone in the shape of a solid disk with a

diameter of 1.0 m and a mass of 50.0 kg, is rotating at 900

rev/min. A tool is pressed against the rim with a normal

force of 200.0 N, and the grindstone comes to rest in 10.0

s. Find the coefficient of friction between the tool and the

grindstone. Neglect friction in the bearings. 0.589

49. A bucket of water of mass 20.0 kg is suspended by a

rope wrapped around a windlass in the form of a solid

cylinder 0.20 m in diameter, also of mass 20.0 kg. The

cylinder is pivoted on a frictionless axle through its

center. The bucket is released from rest at the top of a

well and falls 20.0 m to the water. Neglect the weight of

the rope.

a) What is the tension in the rope while the bucket is

falling? 65.3N

b) With what velocity does the bucket strike the water?

16.2 m/s

c) What was the time of fall? 2.48 sec

d) While the bucket is falling, what is the force exerted

on the cylinder by the axle? 261 N

50. A 60.0-kg grindstone is 1.0 m in diameter and has a

moment of inertia of 3.75 kg-m2. A tool is pressed down

on the rim with a normal force of 50.0 N. The coefficient

of sliding friction between the tool and the stone is 0.6,

and there is a constant friction torque of 5 m-N between

the axle of the stone and its bearings.

a) How much force must be applied normally at the end

of a crank handle 0.5 m long to bring the stone from rest

to120 rev/min in 9.0 s? 50.5 N

b) After attaining a speed of 120 rpm, what must the

normal force at the end of the handle become to maintain

a constant speed of 120 rpm? 40.0 N

c) How long will it take the grindstone to come from 120

rpm to rest if it is acted on by the axle friction alone?

9.42 s

51. Dirk the Dragonslayer is exploring a castle. He is

spotted by a dragon who chases him down a hallway.

Dirk runs into a room and attempts to swing the heavy

door shut before the dragon gets him. The door is initially

perpendicular to the wall, so it must be turned through 90º

to close. The door is 3.00 m tall and 1.00 m wide and

weighs 600.0 N. The friction at the hinges can be

neglected. If Dirk applies a force of 180.0 N at the edge

of the door and perpendicular to it, how long will it take

him to close the door? 0.597 sec

52. A 5.0 kg block rests on

a frictionless horizontal

surface. A cord attached to

the block passes over a pulley

whose diameter is 0.2 m, to

a hanging block also of mass

5.0 kg. The system is

released from rest, and the

blocks are observed to move 4.0 m in 2.0 seconds. a)

What is the tension in each part of the cord?

10 N 39N

b) What is the moment of inertia of the pulley?

0.145 kg-m2

53. Two blocks, one of mass 4.0 kg

and the other of mass 2.0 kg are

connected by a light rope that passes

over a pulley as shown in the figure

to the right. The pulley has radius

0.10 m and moment of inertia

0.20 kg-m2. Find the linear

accelerations of Blocks A and B,

the angular acceleration of wheel C, and the tension in

each side of the cord

a) if the surface of the wheel is frictionless;

aA = aB = 3.27 m/s2; aC = 0; TA = TB = 26.1 N

b) if there is no slipping between the cord and the surface

of the wheel.

aA = aB = 0.745 m/s2; aC = 7.45 rad/s2;

TA = 36.2 N; TB = 21.1 N

54. A block of mass

m = 5 kg slides down a

surface inclined 37º to the

horizontal, as shown in the

figure to the right. The

coefficient of sliding friction

is 0.25. A string attached to

the block is wrapped around a flywheel on a fixed axis at

O. The flywheel has a mass of 20.0 kg, and outer radius

of 0.2 m, and a moment of inertia with respect to the axis

of 0.2 kg-m2.

a) What is the acceleration of the block down the plane?

1.97 m/s2

b) What is the tension in the string? 9.85 N

55. A flywheel 1.0 m in diameter is pivoted on a

horizontal axis. A rope is wrapped around the outside of

the flywheel, and a steady pull of 50.0 N is exerted on the

rope. Ten meters of rope are unwound in 4.0 s.

a) What is the angular acceleration of the flywheel?

b) What is its final angular velocity?

2.5 rad/s2 10 rad/s

c) What is its final kinetic energy? 500 J

d) What is its moment of inertia? 10 kg-m2

Heavy Pulleys and Hanging Masses:

1. A 4.0 kg bicycle wheel

(Mass is concentrated at the rim.)

of radius 0.20 m is held on a

fixed support, while a 1.1 kg

mass on a string wrapped around

the wheel falls as shown. What

is the linear acceleration of the

dropping mass? 2.11 m/s2

2. An Atwood machine is

constructed using a massive 2.0 kg

hoop of 22 cm radius as shown in

the diagram. A 1.5 kg mass and

a 1.0 kg mass arranged as shown

are released from rest. Find the

linear acceleration of the falling

mass. 1.09 m/s2

3. A bicycle wheel of radius

0.70 m and mass 3.0 kg has a small

light hub of radius 0.13 m as

shown in the figure. The 2.0 kg

mass which is attached to a string

wrapped around the hub is released

from rest. What is the linear

acceleration of the dropping mass?

0.220 m/s2

4. An Atwood machine is constructed using two wheels

(Mass concentrated at the rim.) as shown in the figure

below. What is the linear acceleration of the hanging

masses?

5. Find the linear acceleration of the system shown in the

figure below. The mass of the pulley is concentrated at

the rim. The coefficient of kinetic friction between the

ramp and the 5.0 kg block is 0.300. 376 m/s2

6. An Atwood machine consists of

a disk of mass M, and radius R, and

two masses ml and m2 hanging from

each side as shown in the figure.

Find the linear acceleration of the

system.

7. A 2-disk Atwood machine with radii of 15 cm and 38

cm, has a moment of inertia of 4.0 kg-m2 is shown in the

figure below. Masses of 3.0 kg and 2.0 kg are attached to

strings wrapped around the disks as shown. When

released from rest, what is the linear acceleration of each

mass? a2 = 0.105 m/s2 a1 = 0.265 m/s2

9. A spool (solid cylinder) of radius 27 cm is mounted to

spin about its axis. A string wrapped around it is pulled

with a 5.4 N force, causing the object to spin up at 14

rad/sec2. What is the moment of inertia of the object?

0.104 kg-m2

Angular Momentum and Angular Impulse:

The angular momentum of a rigid body about a

fixed axis is defined two ways:

L = Iω and

For a single particle, the angular momentum relative to

any point would be:

...where m is the mass of the particle, r is the position

vector from the point to the particle and v is the

translational velocity.

The product of the torque and the time interval

during which it acts is called the angular impulse, Jθ. The

angular impulse acting on the body causes a change in

the angular momentum of the body about the same axis.

For a torque that varies with time, the angular impulse is

defined as:

Conservation of angular momentum states that when the

net external torque on a system is zero, the angular

momentum of the system remains constant. This principle

of conservation of angular momentum ranks with the

principles of conservation of linear momentum and

conservation of energy as one of the most fundamental of

physical laws.

19. Calculate the angular momentum of a uniform sphere

of radius 0.20 m and mass 4.0 kg if it is rotating about an

axis along a diameter at (a) 6.0 rad/s and (b) 5.0 rev/s.

2.0 kg-m2/sec

20. A solid wooden door 1.0 m wide and 2.0 m high is

hinged along one side and has a total mass of 50.0 kg.

Initially open and at rest, the door is struck at its center

with a hammer. During the blow and average force of

2000.0 N acts for 0.01 seconds. Find the angular velocity

of the door after the impact.

21. A man of mass 70.0 kg is standing on the rim of a

large disk that is rotating at 0.5 rev/s about an axis

through its center. The disk has mass 120.0 kg and radius

4.0 m. Calculate the total angular momentum of the man-

plus-disk system.

22. The outstretched arms of a figure skater preparing for

a spin can be considered a slender rod pivoting about an

axis through its center. When her arms are brought in and

wrapped around her body to execute the spin, they can be

considered a thin-walled hollow cylinder. If her original

angular velocity is 6.28 rad/s, what is her final angular

velocity? Her arms have a combined mass of

8.0 kg. When outstretched they span 1.8 m; when

wrapped, they form a cylinder of radius 25 cm. (A disk

and a cylinder rotating about an axis through the center

have the same moment of inertia.) The moment of inertia

of the remainder of her body is constant and equal to 3.0

kg-m2.

9.26 rad/s if arms considered hollow cylinder

Equilibrium of Rigid Body: Recall that we said the first condition for

equilibrium existed when the sum of the forces acting on

the body was zero. Now we introduce the second

condition for equilibrium which exists when the sum of

the torques of all the forces acting on the body, with

respect to any specified axis is zero. This means that the

body is not accelerating and it is not rotating. If it were

rotating then it would experience a centripetal

acceleration.

21. A 200.0 N weight is hung on the end of a horizontal

pole 2.0 m long. What is the torque around the other end

of the pole caused by this weight? Around the center of

the pole? 400 mN 200 mN

22. Two men carry a 1500 N load by hanging it from a

horizontal pole that rests on one shoulder of each man. If

the men are 3.00 m apart and the load is 1.00 m from one

of them, how much load does each man support? The

weight of the pole is 500 N. 1250 N 750 N

23. A man holds a 2.000 m fishing pole horizontally with

both hands, one at the end and the other 0.300 m from the

end. He has just caught a 1.500 kg fish. The pole has a

mass of 1.000 kg and you can consider its weight to be

concentrated 0.600 m from the end near the man's hands.

What is the force exerted by each hand? 93 N down 118

N up

24. A steel beam of uniform cross section weighs

2.5 x105 N. If it is 5.00 m long, what force is needed to

lift one end of it? 1.2 x 105 N

25. A bar 4.0 m long weighs 400.0 N. Its center of

gravity is 1.5 m from one end. A weight of 300.0 N is

attached at the heavy end and a weight of 500.0 N is

attached at the light end. What are the magnitude,

direction, and point of application of the force needed to

achieve translational and rotational equilibrium of the

bar? 1200.0 N up at 2.2 m from 300 N

26. A painter weighing 875 N stands on a plank 3.00 m

long, which is supported at each end by a stepladder. The

plank weighs 223 N. If the painter stands 1.00 m from

one end of the plank, what force is exerted by each

stepladder? 400 N 700 N

27. A brick layer weighing 800.0 N stands 1.00 m from

one end of a scaffold 3.00 m long. The scaffold

weighs750 N. A pile of bricks weighing 320.0 N is 1.50

m from the other end of the scaffold. What force must be

exerted on each end of the scaffold in order to support it?

1070 N at end near bricklayer 800 N other end

Conceptual Questions: 1. Does a record player needle ride faster or slower over the

groove at the beginning or the end of the record? If fidelity

increases with translational speed, what part of the record

produces the highest fidelity?

2. Suppose the first and last selections on a phonograph record

are 3-minutes cuts. Which, if either, of these cuts is wider on

the record? (That is, which contains more grooves along a radial

direction?)

3. Which moves faster on a merry-go-round, a horse near the

center or one near the outside.

4. If you use large diameter tires on your car, how will your

speedometer reading differ?

5. Why are the front wheels located so far out in front on the

racing vehicle?

6. Which will roll down a hill faster, a cylinder or a sphere of

equal radii? A hollow cylinder or a solid cylinder of equal radii?

Explain.

7. Why do buses and heavy trucks have large steering wheels?

8. Which is easier for turning stubborn screws, a screwdriver

with a thick handle or one with a long handle? Explain.

9. Why is the middle seating most comfortable in a bus

traveling on a bumpy road?

10. Explain why a long pole is more beneficial to a tightrope

walker if it droops.

11. Why do you bend forward when carrying a heavy load on

your back?

12. Why is it easier to carry the same amount of water in two

buckets, one in each hand, then in a single bucket?

13. Using the ideas of torque and center of gravity, explain why

a ball rolls down a hill.

14. Why is it dangerous to roll open the top drawers of a fully

loaded file cabinet that is not secured to the floor?

15. Why is less effort required in doing sit-ups when your arms

are extended in front of you? Why is it more difficult when

your arms are placed in back of your head?

16. For a rotating wheel, how do the directions of the linear

velocity vector and the angular velocity vector compare at the

same instant of time?

Answers to conceptual questions:

1. The phonograph needle rides faster at the beginning of the

record. Since fidelity is enhanced with translational speed, then

fidelity would be best at the beginning of a record.

2. Both three minute selections would have the same width

because they would make the same number of revolutions

during a three minute time period.

3. The horse on the outer rail has a greater translational

(tangential) speed, while both have the same rotational speed.

4. The circumference of a large diameter tire is greater,

meaning it will move a greater distance per revolution, which

results in a greater speed than that shown on the speedometer.

5. The long distance to the front wheels increases the rotational

inertia of the vehicle without appreciably adding to its weight.

As the back wheels are driven clockwise, the rest of the car

tends to rotate counter-clockwise. This would lift the front

wheels off the ground.

6. A sphere will roll faster because it has less rotational inertia

than a cylinder. A solid cylinder will roll faster than a hollow

cylinder for the same reason.

7. The large radius of a large steering wheel allows the driver to

exert more torque for a given force.

8. More torque can be exerted by the screw driver having a

thick handle.

9. A rocking bus rocks about its center of gravity which is

around the center of the bus. It works something like a see-saw

- the farther from the center, the more you go up and down.

10. The long drooping pole lowers the center of gravity of the

pole and the tightrope walker. The pole contributes to his

rotational inertia.

11. You bend forward to shift the center of gravity of you and

the back pack. If you did not shift the center of gravity over the

support, you would topple over.

12. There is no need to adjust your center of gravity if the water

is distributed between the two buckets.

13. When a ball is on an incline its center of gravity is not

above the point of support. The weight acts some distance from

the point of support and produces a torque about the point of

support.

14. The center of gravity could be adjusted so that it is no

longer above the support.

15. When your arms are extended in front of you while doing

sit-ups, not only are they not lifted as far, they are closer to the

axis of rotation and give you less rotational inertia. When

behind your head they are lifted farther and their farther distance

from the axis of rotation increases your rotational inertia.