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Chapter 6 Angular Motion

Objectives After completing this module, you

should be able to

- Define and apply concepts of angular

displacement, velocity, and acceleration. - Draw analogies relating rotational-motion

parameters (?, ?, ?) to linear (x, v, a) and

solve rotational problems. - Write and apply relationships between linear and

angular parameters.

Objectives (Continued)

- Define moment of inertia and apply it for several

regular objects in rotation. - Apply the following concepts to rotation
- 1. Rotational work, energy, and power
- 2. Rotational kinetic energy and momentum
- 3. Conservation of angular momentum

Rotation An Introduction

- For some to travel once around a circle, the

distance formula is 2pr - The time to go once around is referred to as the

Period (T) - V
- In a circle V

Rotational Displacement, ?

Consider a disk that rotates from A to B

B

Angular displacement q

?

Measured in revolutions, degrees, or radians.

1 rev 360 0 2? rad

The best measure for rotation of rigid bodies is

the radian.

Definition of the Radian

One radian is the angle ? subtended at the center

of a circle by an arc length s equal to the

radius R of the circle.

360o 2

rad

Definition of the Radian

rad 57.3o

Converting rads to degrees and degrees to rads

Example 1 A rope is wrapped many times around a

drum of radius 50 cm. How many revolutions of

the drum are required to raise a bucket to a

height of 20 m?

q 40 rad

Now, 1 rev 2p rad

q 6.37 rev

Example 2 A bicycle tire has a radius of 25 cm.

If the wheel makes 400 rev, how far will the bike

have traveled?

q 2513 rad

s q R 2513 rad (0.25 m)

s 628 m

Angular Velocity

Angular velocity,w, is the rate of change in

angular displacement. (radians per second.)

Angular velocity,?, can also be given as the

frequency of revolution, f (rev/s or rpm)

w 2pf rad/s

Example 3 A rope is wrapped many times around a

drum of radius 20 cm. What is the angular

velocity of the drum if it lifts the bucket to 10

m in 5 s?

q 50 rad

w 10.0 rad/s

Example 4 In the previous example, what is the

frequency (rev/s) of revolution for the drum?

Recall that w 10.0 rad/s.

Or, since 60 s 1 min

f 95.5 rpm

Angular Velocity continued

Angular velocity,w, can also be written in terms

of the number of revolutions (2 r) per time of

one turn.

2 rad T

w Angular velocity in rad/s.

Angular velocity,?, can also be given as the

frequency of revolution, f (rev/s or rpm)

w (2p rad) f

Linear Speed

Linear speed, can be written in terms of the

circumference (2pr) per time of one turn.

v w r speed (m/s)

Angular and Linear Speed

From the definition of angular displacement

s q R Linear vs. angular displacement

Linear speed angular speed x radius

Examples

Consider flat rotating disk

wo 0 wf 20 rad/s

R1 20 cm R2 40 cm

What is final linear speed at points A and B?

vAf wAf R1 (20 rad/s)(0.2 m) vAf 4 m/s

vAf wBf R2 (20 rad/s)(0.4 m) vBf 8 m/s

Angular Acceleration

v2

a

2

Regular acceleration m/s

r

Angular acceleration is the rate of change in

angular velocity. (Radians per sec per sec.)

The angular acceleration can by subsituting wr

for v and you would get (w2r2)/r OR

a w2r m/s2

Angular Acceleration

Angular acceleration is the rate of change in

angular velocity. (Radians per sec per sec.)

The angular acceleration can also be found from

the change in frequency, as follows

Acceleration Example

What is the average angular and linear

acceleration at B?

a 5.00 rad/s2

a 2.00 m/s2

a aR (5 rad/s2)(0.4 m)

Example 5 The block is lifted from rest until

the angular velocity of the drum is 16 rad/s

after a time of 4 s. What is the average angular

acceleration?

a 4.00 rad/s2

Angular and Linear Acceleration

From the velocity relationship we have

v wR Linear vs. angular velocity

Linear accel. angular accel. x radius

Angular vs. Linear Parameters

But, a aR and v wR, so that we may write

6.3 Dynamics (Forces) and Circular Motion

Known Equations Fnet m x a

6.3 Dynamics (Forces) and Circular Motion

An 80 kg person steps into the graviton 2000 that

has a radius of 8 m. Graviton 2000 makes 15

turns per minute. What linear velocity does the

person have? With what force does the wall push

on the person

2pr T

2p 8m 4 s

v

v

12.57 m/s

v

6.3 Dynamics (Forces) and Circular Motion

An 80 kg person steps into the graviton 2000 that

has a radius of 8 m. Graviton 2000 makes 30

turns per minute. What linear velocity does the

person have? With what force does the wall push

on the person

mv2 r

(80kg) (12.57 m/s)2 8m

Fnet

Fnet

Fnet

1579N

6.4 Apparent Forces during Circular Motion

The circular Fnet equation works well to

determine the net force, but it doesnt determine

how much force the wall or string applies if

there are outside forces. If you ever gone

through a loop in a roller coaster, you know that

you feel heavier at the bottom than at the top

when you are upside down.

mv2 r

n

Fnet

w

Fnet n - w

6.5 Orbits

Earth

6.5 Orbits

A look back at Projectile Motion objects are

given only horizontal velocity

Earth

6.5 Orbits

Zoom out and move the projectile up 3,000 or so

miles

6.5 Orbits

Zoom out again about 25,000 miles

Earth

Which way is gravity when the object is here?

Which way is gravity when the object is here?

6.5 Orbits

So how fast would an object need to go to orbit??

v2 r

v2/r since orbit is circular

a g

v2 r g

At the surface of the earth, g 9.8 m/s/s

At the surface of the earth, radius 6.37 x 106 m

6.5 Orbits

So how fast would an object need to go to orbit??

v

v (6.37 x 106 m)( 9.8m/s2)

v

v (6.24 x 107 m)

v 7900 m/s about 5 miles/ second!!!

Problem the acceleration of gravity from the

earth is not always 9.8m/s2 it decreases as you

get farther away

6.6 Newtons Law of Gravity

- For every force there is an equal and opposite

force - Forces is are inversely proportional to the

square of the distance between them - The force is directly proportional to the

product of the masses of the two objects.

6.6 Newtons Law of Gravity

Fgravity E on M

Fgravity M on E

Earth

Moon

G m1 m2 r2

F(M on E) F(E on M)

Fgravity

G (Gravitational Constant) 6.67 x 10-11 N

m2/kg2

r is the radius or distance between the centers

of the objects in this case the moon and the

earth.

Linear Example A car traveling initially at 20

m/s comes to a stop in a distance of 100 m. What

was the acceleration?

a -2.00 m/s2

Angular analogy A disk (R 50 cm), rotating at

600 rev/min comes to a stop after making 50 rev.

What is the acceleration?

wo 600 rpm wf 0 rpm q 50 rev

50 rev 314 rad

a -6.29 m/s2

Problem Solving Strategy

- Draw and label sketch of problem.
- Indicate direction of rotation.
- List givens and state what is to be found.

Given ____, _____, _____ (q,wo,wf,a,t) Find

____, _____

- Select equation containing one and not the other

of the unknown quantities, and solve for the

unknown.

Example 6 A drum is rotating clockwise

initially at 100 rpm and undergoes a constant

counterclockwise acceleration of 3 rad/s2 for

2 s. What is the angular displacement?

Given wo -100 rpm t 2 s a 2 rad/s2

q -14.9 rad

q -20.9 rad 6 rad

Net displacement is clockwise (-)

Summary of Formulas for Rotation

CONCLUSION Chapter 11A Angular Motion