Rotational Motion 1

Circular Motion

Rotational Motion 2

Uniform Circular Motion

An object that moves in a circle at constant speed issaid to experience uniform circular motion.

• The magnitude of the velocity remains constant.• The direction of the velocity is continuously

changing as the object moves around the circle.• The object is accelerating because there is a

change in velocity.

This acceleration is called centripetal accelerationand it points towards the center of the circle.

Rotational Motion 3

Centripetal Acceleration

v1v2

!v

F

E

D

v1

v2

A

B

C

r

!

arad ="v"t

!

"DEF # "ABC

!

"v = v2 # v1

!

"vv

=ABr

!

so "vv

!

or v2 = "v + v1

!

and AB = d

!

= v " #t

!

= ac

!

=v 2

r

!

=v 2 " #tr " #t

!

and "v"t

!

=v " #tr

Rotational Motion 4

Centripetal Acceleration

• This component always points towards the axis of rotation.

• The centripetal acceleration is always perpendicularto tangential motion.

!

v a c

!

v v

!

v a c

!

v v

!

v v

!

v a c

!

v a c

!

v v

!

v a c

!

v v

!

v a c

!

v v

!

v a c

!

v v

!

v a c

!

v v

!

ac =v2

r

Rotational Motion 5

Forces in Circular Motion

Because an object in uniform circular motion isaccelerating, there must be a net force creating thisacceleration. Therefore, Newton’s second-law canbe applied to problems involving circular motion.This net force is called a centripetal force whichcauses the centripetal acceleration.

!

Fr = m" ac

!

= mv2

rwhere is the sum of all forces in the radialdirection (towards or away from the center of thecircle).

!

Fr"

Rotational Motion 6

Forces in Circular Motion

Examples of forces that result in circular motioninclude:• Tensions in cords swinging objects in circular

paths.• Normal forces on objects in motion on roller

coaster loops and Ferris wheels.• Frictional forces on objects moving on curved

roads.• Gravitational forces between objects orbiting

other objects.

7

Vertical Motion of a Mass on a Cord

Fg

Fg

FgFg

Top

Bottom

m

m

m

mac

ac

ac

ac

!

v F net = mv a

!

T " Fg =mac

!

T =mac + Fg

!

T =m ac + g( )

T

T

!

v F net = mv a

!

T + Fg =mac

!

T =mac " Fg

!

T =m ac " g( )

Rotational Motion 8

Roller Coaster Loops

!

v F net = mv a

!

FN " Fg =mac

!

FN = Fg +mac

!

FN =mg +mac

!

FN =m g + ac( )!

FN

!

Fg!

ac

!

FN

!

Fg

!

ac

!

v F net = mv a

!

Fg + FN =mac

!

FN =mac " Fg

!

FN =mac "mg

!

FN =m ac " g( )

Rotational Motion 9

Ferris Wheel

!

v F net = mv a

!

Fg " FN =mac

!

FN = Fg "mac

!

FN =mg "mac

!

FN =m g " ac( )!

FN

!

Fg

!

ac

!

FN

!

Fg!

ac

!

v F net = mv a

!

FN " Fg =mac

!

FN = Fg +mac

!

FN =mg +mac

!

FN =m g + ac( ) Rotational Motion 10

Car on a Curved Road

!

ac

!

Ff

!

Ff

!

Ff!

FN

!

Fg!

m

(View from above) (Side view)

!

v F net = mv a

!

Ff =mac

!

µFN =mac

!

µmg =mac

!

µg = ac

!

F" y =ma

!

FN " Fg = 0

!

FN = Fg =mg

!

F" x =ma