Rotational Motion and Angular Momentum

- Unit 6

Lesson 1 Angular Position, Velocity, and

Acceleration

When a rigid object rotates about its axis, at

any given time different parts of the object have

different linear velocities and linear

accelerations.

P is at polar coordinate (r, q)

s rq

q is measured in radians (rad)

Converting from Degrees to Radians

90o p/2 rad

60o p/3 rad

45o p/4 rad

270o 3p/2 rad

Angular Displacement (Dq)

Average Angular Speed (w)

Ratio of the angular displacement of a rigid

object to the time interval Dt.

The rad/s is the unit for angular speed.

w is positive when q increases (counterclockwise

motion)

w is negative when q decreases (clockwise motion)

Instantaneous Angular Speed (w)

Instantaneous Angular Acceleration (a)

The rad/s2 is the unit for angular speed.

a is positive when object rotates

counterclockwise and speeds up

OR

when object rotates clockwise and slows down

Direction of Angular Velocity and Acceleration

Vectors

w and a are vector quantities with magnitude and

direction

Right-Hand Rule

Wrap four fingers of the right hand in the

direction of rotation.

Thumb will point in the direction of angular

velocity vector (w).

Example 1

A rigid object is rotating with an angular speed

w lt 0. The angular velocity vector w and the

angular acceleration vector a are antiparallel.

The angular speed of the rigid object is

a) clockwise and increasing

b) clockwise and decreasing

c) counterclockwise and increasing

d) counterclockwise and decreasing

Example 2

During a certain period of time, the angular

position of a swinging door is described by q

5.00 10.0 t 2.00 t2, where q is in radians

and t is in seconds. Determine the angular

position, angular speed, and angular acceleration

of the door at

a) at t 0

b) at t 3.00 s

Lesson 2 Rotational Kinematics with Constant

Angular Acceleration

dw a dt

Eliminating t from previous two equations,

Eliminating a from previous two equations,

x

Position

q

v

Velocity

w

a

Acceleration

a

Example 1

A wheel rotates with a constant angular

acceleration of 3.50 rad/s2.

a) If the angular speed of the wheel is 2.00

rad/s at ti 0, through what angular

displacement does the wheel rotate in 2.00 s ?

b) Through how many revolutions has the wheel

turned during this time interval ?

c) What is the angular speed of the wheel at t

2.00 s ?

Example 2

A rotating wheel requires 3.00 s to rotate

through 37.0 revolutions. Its angular speed at

the end of the 3.00 s interval is 98.0 rad/s.

What is the constant angular acceleration of the

wheel ?

Lesson 3 Angular and Linear Quantities

Since s rq,

Tangential Speed

v rw

Tangential speed depends on distance from axis of

rotation

Angular speed is the same for all points

Tangential Acceleration

Since v rw,

Centripetal Acceleration in terms of Angular Speed

Since v rw,

Total Linear Acceleration

Example 1

a) Find the angular speed of the disc in rev/min

when information is being read from the

innermost first track (r 23 mm) and the

outermost final track (r 58 mm).

b) The maximum playing time of a standard music

CD is 74 min 33 s. How many revolutions does

the disc make during that time ?

c) What total length of track moves past the lens

during this time ?

d) What is the angular acceleration of the CD

over the 4,473 s time interval ? Assume that a

is constant.

Example 2

a) Calculate the speed of a link of the chain

relative to the bicycle frame.

b) Calculate the angular speed of the bicycle

wheels.

c) Calculate the speed of the bicycle relative

to the road.

d) What pieces of data, if any, are not

necessary for the calculations ?

Lesson 4 Rotational Kinetic Energy

The total kinetic energy of a rotating rigid

object is the sum of the kinetic energies of its

individual particles.

KErot SKEi S ½mivi2

Since v rw,

KErot ½ S miri2wi2

Factoring out w2,

KErot ½ (S miri2) w2

KErot ½ (S miri2) w2

The kg . m2 is the SI unit for moment of inertia.

Substituting I,

I

m

w

v

Example 1

Consider an oxygen molecule (O2) rotating in the

x-y plane about the z-axis. The rotation axis

passes through the center of the molecule,

perpendicular to its length. The mass of each

oxygen atom is 2.66 x 10-26 kg, and at room

temperature the average separation between the

two atoms is d 1.21 x 10-10 m. (The atoms are

modeled as particles.)

a) Calculate the moment of inertia of the

molecule about the z-axis.

b) If the angular speed of the molecule about the

z-axis is 4.60 x 1012 rad/s, what is its

rotational kinetic energy ?

Lesson 5 Calculation of Moments of Inertia

Moment of inertia of a rigid object is evaluated

by dividing the object into many small volume

elements, each with mass Dmi.

Since r m/V,

dm r dV

Example 1

Example 2

Example 3

Parallel-Axis Theorem

Example 4

Moment of Inertia of a Thin Cylindrical Shell

(Hoop)

Moment of Inertia of a Hollow Cylinder

Moment of Inertia of a Solid Cylinder (Disk)

Moment of Inertia of a Rectangular Plate

Moment of Inertia of a Long Thin Rod

(Axis Through Center)

Moment of Inertia of a Long Thin Rod

(Axis Through End)

Moment of Inertia of a Solid Sphere

Moment of Inertia of a Thin Spherical Shell

Lesson 6 Torque

The tendency of a force to rotate an object about

some axis is measured by a vector quantity called

torque.

t r Fsinf Fd

Torque has units of force x length or N.m. (Same

as work but not called Joules.)

F1 tends to rotate counterclockwise (t)

F2 tends to rotate clockwise (-t)

Example 1

a) What is the net torque acting on the cylinder

about the rotation axis (z-axis) ?

b) Suppose T1 5.0 N, R1 1.0 m, T2 15.0 N,

and R2 0.50 m. What is the net torque about

the rotation axis, and which way does the

cylinder rotate starting from rest ?

Lesson 7 Relationship between Torque and

Angular Acceleration

Ft mat

t Ftr (mat)r

t (mra)r (mr2)a

Since I mr2,

Example 1

Example 2

a) Calculate the angular acceleration of the

wheel.

b) Calculate the linear acceleration of the

object.

c) Calculate the tension in the cord.

Lesson 8 Work, Power, and Energy in Rotational

Motion

dW F.ds (Fsinf)r dq

(The radial component of F does no work because

it is perpendicular to the displacement.)

Since t rFsinf,

St dq Iw dw

Since dW St dq,

dW Iw dw

Integrating to find total work done,

The net work done by external forces in rotating

a symmetric rigid object about a fixed axis

equals the change in the objects rotational

energy.

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Example 1

a) What is its angular speed when it reaches its

lowest position ?

b) Determine the tangential speed of the center

of mass and the tangential speed of the lowest

point on the rod when it is in the vertical

position.

Example 2

Example 3 AP 2001 3

a) Determine the rotational inertia of the

rod-and- block apparatus attached to the top of

the pole.

b) Determine the downward acceleration of the

large block.

c) When the large block has descended a distance

D, how does the instantaneous total kinetic

energy of the three blocks compare with the

value 4mgD ? Check the appropriate space below.

____ Greater than 4mgD

____ Equal to 4mgD

____ Less than 4mgD

Justify your answer.

The system is now reset. The string is rewound

around the pole to bring the large block back to

its original location. The small blocks are

detached from the rod and then suspended from

each end of the rod, using strings of length l.

The system is again released from rest so that as

the large block descends and the apparatus

rotates, the small blocks swing outward, as shown

in Experiment B above. This time the downward

acceleration of the block decreases with time

after the system is released.

d) When the large block has descended a distance

D, how does the instantaneous total kinetic

energy of the three blocks compare to that in

part c) ? Check the appropriate space below.

____ Greater

____ Equal

____ Less

Justify your answer.

Lesson 9 Rolling Motion of a Rigid Object

Center moves in a straight line (green line).

A point on the rim moves in a path called a

cycloid (red curve).

Speed of CM of Cylinder Rolling without Slipping

Since s Rq,

Acceleration of CM of Cylinder Rolling without

Slipping

Total Kinetic Energy of a Rolling Cylinder

Since IP ICM MR2, (parallel-axis theorem)

KErot ½ (ICM MR2)w2

Since vCM Rw,

KEtotal ½ ICMw2 ½ MvCM2

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Example 1

Example 2

Example 3 AP 1986 2

a) Determine the following for the sphere when it

is at the bottom of the plane.

i. Its translational kinetic energy

ii. Its rotational kinetic energy

b) Determine the following for the sphere when it

is on the plane.

i. Its linear acceleration

ii. The magnitude of the frictional force acting

on it

The solid sphere is replaced by a hollow sphere

of identical radius R and mass M. The hollow

sphere, which is released from the same location

as the solid sphere, rolls down the incline

without slipping.

c) What is the total kinetic energy of the hollow

sphere at the bottom of the plane

d) State whether the rotational kinetic energy of

the hollow sphere is greater than, less than,

or equal to that of the solid sphere at the

bottom of the plane. Justify your answer.

Lesson 10 Angular Momentum

dr

dr

x p is zero since v,

dt

dt

and v and p are parallel.

Instantaneous Angular Momentum

The instantaneous angular momentum L of a

particle relative to the origin O is defined by

the cross product of the particles instantaneous

position vector r and its instantaneous linear

momentum p.

The SI unit of angular momentum is kg . m2/s.

Since L r x p,

The torque acting on a particle is equal to the

time rate of change of the particles angular

momentum.

The direction of L is always perpendicular to the

plane formed by r and p.

Example 1

Angular Momentum of a System of Particles

The total angular momentum of a system of

particles about some point is defined as the

vector sum of the angular momenta of the

individual particles.

Differentiating with respect to time

The net external torque acting on a system about

some axis passing through an origin in an

inertial frame equals the time rate of change of

the total angular momentum of the system about

that origin.

This theorem applies even if the center of

mass is accelerating, as long as t and L are

evaluated relative to the center of mass.

Example 2

Lesson 11 Angular Momentum of a Rotating Rigid

Object

Since v rw,

L mr2w

Since I mr2,

L Iw

Example 1

Example 2

a) Find an expression for the magnitude of the

systems angular momentum.

b) Find an expression for the magnitude of the

angular acceleration of the system when the

seesaw makes an angle q with the horizontal.

Example 3 AP 1983 2

a) On the diagram below draw and identify all of

the forces acting on the cylinder and on the

block.

b) In terms of m1, m2, R, and g, determine each

of the following.

i) The acceleration of the block.

ii) The tension in the cord.

iii) The angular momentum of the disk as a

function of t.

Example 4 AP 1982 3

a) The initial angular momentum of the system

about the axis through P.

b) The frictional torque acting on the system

about the axis through P.

c) The time T at which the system will come to

rest.

Example 5 AP 1996 3

a) Show that the rotational inertia of the rod

about an axis through its center and

perpendicular to its length is Ml 2/12.

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b) What is the rotational inertia of the rod-hoop

assembly about the axle ?

Several turns of string are wrapped tightly

around the circumference of the hoop. The system

is at rest when a cat, also of mass M, grabs the

free end of the string and hangs vertically from

it without swinging as it unwinds, causing the

rod-hoop assembly to rotate. Neglect friction and

the mass of the string.

c) Determine the tension T in the string.

d) Determine the angular acceleration a of the

rod-hoop assembly.

e) Determine the linear acceleration of the cat.

f) After descending a distance H 5 l /3, the

cat lets go of the string. At that instant, what

is the angular momentum of the cat about point P

?

Lesson 12 Conservation of Angular Momentum

The total angular momentum of a system is

constant in both magnitude and direction if the

resultant external torque acting on the system is

zero, that is, if the system is isolated.

Lbefore Lafter

Since L Iw,

Example 1

Example 2

Example 3

Example 4 AP 1987 3

Immediately after the collision the object moves

with speed v at an angle q relative to its

original direction. The bar swings freely, and

after the collision reaches a maximum angle of

90o with respect to the vertical. The moment of

inertia of the bar about the pivot is Ibar ml

2/3. Ignore all friction.

a) Determine the angular velocity of the bar

immediately after the collision.

b) Determine the speed v of the 1 kg object

immediately after the collision.

c) Determine the magnitude of the angular

momentum of the object about the pivot just

before the collision.

d) Determine the angle q.

Example 5 AP 1992 2

a) Determine the torque about the axis

immediately after the bug lands on the sphere.

b) Determine the angular acceleration of the

rod- spheres-bug system immediately after the bug

lands.

The rod-spheres-bug system swings about the axis.

At the instant that the rod is vertical, as shown

above, determine each of the following.

c) The angular speed of the bug.

d) The angular momentum of the system.

e) The magnitude and direction of the force that

must be exerted on the bug by the sphere to

keep the bug from being thrown off the sphere.

Lesson 13 Rotational Equilibrium

A system is in rotational equilibrium if the net

torque on it is zero about any axis.

Since St Ia,

St 0 does not mean an absence of rotational

motion. Object can be rotating at a constant

angular speed.

Example 1

____ force equilibrium but not torque equilibrium.

____ torque equilibrium but not force equilibrium.

____ both force and torque equilibrium.

____ neither force nor torque equilibrium.

Example 2

____ force equilibrium but not torque equilibrium.

____ torque equilibrium but not force equilibrium.

____ both force and torque equilibrium.

____ neither force nor torque equilibrium.

Center of Gravity

To compute the torque due to the gravitational

force on an object of mass M, we need only

consider the force Mg acting at the center of

gravity of the object.

Center of gravity center of mass if g is

constant over the object.

Example 3

a) Determine the magnitude of the upward force n

exerted by the support on the board.

b) Determine where the father should sit to

balance the system.

Example 4