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Circular Motion

Acceleration and why weird

things happen to spinning items

As an object moves around a circle, its

direction of motion is constantly

changing.

Therefore its velocity is changing.

Therefore an object moving in a circle is

constantly accelerating.

Recall Angular VelocityRecall that Angular velocity, is the rate of change in

angular displacement. (radians per second.)

Angular velocity in rad/s.t

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 (remember s is arc length

and is angular displacement) in 5 s?

h = 10 m

R

= 10.0 rad/s

t50 rad

5 s

= 50 rad10 m

0.20 m

s

R

Angular AccelerationAngular acceleration is the rate of change in angular velocity.

(Radians per sec per sec.)

2 Angular acceleration (rad/s )t

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?

h = 20 m

R

= 4.00 rad/s2

0

f o f

ort t

2

16 rad/s rad4.00

4 s s

2 Angular acceleration (rad/s )t

Angular and Linear

Acceleration:• Linear Acceleration (Sometimes called Tangential

Acceleration) – acceleration at a radius. How fast

something is speeding up in a straight line.

• A runner in the outside lane has to have more tangential

acceleration to stay along a runner in the inside lane.

a = r

Linear acceleration = angular acceleration x radius

Examples:

R1 = 20 cm R2 =

40 cm

R1

R2

A

B

= 0; f = 20 rad/s

t = 4 s

What is final linear speed at points

A and B?

Consider flat rotating disk:

vAf = R1 Af = (0.2 m)(20 rad/s); vAf = 4 m/s

vAf = R1 Bf = (0.4 m)(20 rad/s); vBf = 8 m/s

v = r

Acceleration Example

R1 = 20 cm

R2 = 40 cm

What is the average angular and linear

acceleration at B?

R1

R2

A

B= 0; f = 20 rad/s

t = 4 s

Consider flat rotating disk:

= 5.00 rad/s2

a = R = (5 rad/s2)(0.4 m) a = 2.00 m/s2

0 20 rad/s

4 s

f

t

Centripetal Acceleration

The acceleration of an object moving in

a circle points toward the center of

the circle.

This is called a centripetal (center

pointing) acceleration.

a

Centripetal Acceleration

The centripetal acceleration depends

on:

The speed of the object.

The radius of the circle.

Acent = v2

r

A Comparison: Linear vs. Angular

0

2

fv vs vt t 0

2

ft t

f o tf ov v at

210 2t t21

0 2s v t at

212f t t

2 2

02 f

2 2

02 fas v v

212fs v t at

Centripetal ForceNewton’s Second Law says that if an

object is accelerating, there must be a net

force on it.

For an object moving in a circle, this is

called the centripetal force.

The centripetal force points toward the

center of the circle.

Centripetal Force

In order to make an object revolve

about an axis, the net force on the

object must pull it toward the center

of the circle.

This force is called a centripetal (center

seeking) force.Fnet

Centripetal Force

Centripetal force on an object depends

on:

The object’s mass - more mass means

more force.

The object’s speed - more speed means

more force.

And…

Centripetal Force

The centripetal force on an object also depends on:

The object’s distance from the axis (radius). If linear velocity is held constant, more

distance requires less force.

If rotational velocity is held constant, more distance requires more force.

Centripetal Force

In symbols:

Fcent=mv2

r= mr 2

“Centrifugal Force”

“Centrifugal force” is a fictitious force -

it is not an interaction between 2

objects, and therefore not a real

force.

Nothing pulls an object away from

the center of the circle.

“Centrifugal Force”

What is erroneously attributed to

“centrifugal force” is actually the action

of the object’s inertia - whatever

velocity it has (speed + direction) it

wants to keep.

If you have a bolt that is stuck, do you want to use a short wrench

or a long wrench? Why?

“Torque”

Torque - A measurement of the tendency

of a force to produce a rotation about an

axis. = F x d

is torqueF is the applied force some distinct form the axis.

d is distance from the axis or pivot point

If torque creates a:

Counterclockwise rotation then it is positive

Clockwise rotation then it is negative.

Center of Gravity

Center of gravity – The point on any

object that acts like the place at which

all the weight is concentrated.

The End