Uniform Circular Motion

• Uniform circular

motion is motion

in a circle at the

same speed

• Speed is

constant, velocity

direction

changes

• the speed of an object moving in a circle is given

by

2 r

Tv

circumference

time

r= radius of circle

T = period, time for 1 complete

circle

Moon setting

over Rio de

Janerio

Pictures every

6.5 minutes

showing the

moon is moving

at a uniform

speed

• Since the velocity is changing direction, there is

an acceleration

• the centripetal acceleration of the object is

r

2vc

a

ac = centripetal acceleration

(always to centre of circle)

2

2 2

2

2vac r

2

4

r

T

r

r

Tr

2

2

4 r

T

On data sheet too!

•the velocity vector is

always directed along the

tangent of the circle (at

right angles to the

acceleration vector)

•centripetal means centre-

seeking, the acceleration

vector always points to

the centre of the circle

Centripetal Force

• inertia tends to keep objects moving in a

straight line, a force is needed to cause a

circular motion (2nd Law of Motion)

• the force causing the motion is called the

centripetal force and it always acts

towards the centre of the circle (parallel to

the acceleration vector)

Centrifuge Training

2

F m a & net acv

r

NOT on data sheet

On data

sheet

2mv F c

r

• an object will

move in a circle

whenever a

constant

magnitude force

acts on the object

at right angles

to its direction

of motion

• The force and

acceleration are

radially inward

• The velocity is

tangential

• The string pulls ball

towards the centre of

the circle

• 3rd Law of Motion: a

reaction force will act

outward (ball pulls

string)

• this is sometimes

called centrifugal force

Important!!!

• centripetal force is not a fundamental force

like gravity, it is the net force acting on an

object moving in a curved path

• The force will not change the speed of the

object because the force has NO

component parallel to the velocity

vector

Practice

• A centrifuge for

pilot training is 11

m in radius. At

what speed must it

rotate in order to

inflict 7.0g of

acceleration on a

pilot?

Solution

• 𝑎𝑐 =𝑣2

𝑟

• 𝑣2 = 𝑎𝑐 𝑟

• 𝑣 = 𝑎𝑐𝑟

• 𝑣 = 7 × 9.81𝑚/𝑠2 × 11𝑚

• v = 27 m/s

Frequency of rotation

• sometimes speed is given in revolutions per minute (rpm) or revolutions per second (rps)

• this is how many times the object goes around the circle in 1 minute or 1 second

• frequency is the number of cycles per second measured in hertz (Hz)

11 sondsec

Hz

fT

1

Example

• Find the period of an object that rotates at 35.0

rps.

T

circle

s

circles 1

1

0.35

sT 0286.0

Free Body Diagrams

• A tetherball is attached to a

swivel in the ceiling by a light

cord. When the ball is hit by a

paddle, it swings in a horizontal

circle with constant speed, and

the cord makes a constant

angle with the vertical

direction. Write the expression

for the centripetal force in

terms of the other forces.

Free Body Diagrams

• An object is on a horizontal disc that is

rotating at constant speed. Friction

prevents the rock from sliding. Write the

expression for the centripetal force in

terms of the other forces.

•

Example

• A 3.50 kg object is swung in a 1.50 m radius

horizontal circle at 40.0 rps (40.0 Hz). What

magnitude force acts on the object?

2

2

4 mF m a F net c

T

r

Solution

2

2

4 mF m a F net c

T

r

N 510 x 3.32cF

s) (0.0250

kg) 3.50m) (1.504cF

r4cF

2

(2

2

2

T

m

Solution #2

2

2

4 mF m a F net c

T

r

N 510 x 3.32cF

s) (0.0250

kg) 3.50m) (1.504cF

r4cF

2

(2

2

2

T

m

224F so 1

rmfT

f

Effect of radius on speed and forces

• all points on a rotating

solid have the same

period, but different

speeds

• because the inner points

have smaller distances to

travel, their speeds are

less

• the speed and force

depends on the radius

• NASA and other space agencies use this to help launch satellites

• at the equator, the speed is about 1667 km/h, at Edmonton, it is about 900 km/h

Practice

• A 1100 kg car is moving

in a circle at 15.0 m/s. It

takes the car 12.5 s to

move from position 1 to

position 2. What is the

centripetal acceleration

of the car?

Solution

• Initial velocity = 15.0 m/s

south

• Final velocity = 15.0 m/s

west

if vvv

fv

iv

if vv

if vvv