Circular Motion

Chapter 9 in the TextbookChapter 6 is PSE pg. 81

Revolve: is to move around an EXTERNAL axis

Rotate: is to move around an INTERNAL axis

So the moon _________ around the earth

The earth _________ once a day

The earth _________ once a year

ROTATES

REVOLVES

REVOLVES

The PERIOD (T) of an object is the time it takes the mass to make a complete revolution or rotation.

UNITS:

T in seconds

f in Hz (s-1)

The FREQUENCY (f) of an object is the number of turns per second

fT

1 T = 1

f

Period and frequency are reciprocals

A spring makes 12 vibrations in 40 s. Find the period and frequency of the vibration.

f = vibrations/time = 12vib/40s = 0.30 Hz T = 1/f = 1/(0.3Hz) = 3.33 s

The period T is the time for one complete revolution. So the linear speed or tangential speed can be found by dividing the period into the circumference:

vr

T2

Units: m/s

v = 2π r f

A 2 kg body is tied to the end of a cord and whirled in a horizontal circle of radius 2 m. If the body makes three complete revolutions every second, determine its period and linear speed

m = 2 kgr = 2 mf = 3 rev/s srevf

T/3

11 = 0.33 s

T

rv

2

s

m

33.0

)2(2 = 37.70 m/s

UNIFORM CIRCULAR MOTION

Uniform circular motion is motion in which there is no change in speed, only a change in direction.

CENTRIPETAL ACCELERATION

An object experiencing uniform circular motion is continually accelerating. The direction and velocity of a particle moving in a circular path of radius r are shown at two instants in the figure. The vectors are the same size because the velocity is constant but the changing direction means acceleration is occurring.

r

vac

2

To calculate the centripetal acceleration, we will use the linear velocity and the radius of the circle

Or substituting for v We get

2

24

T

rac Or ac = 4 π2 r f2

A ball is whirled at the end of a string in a horizontal circle 60 cm in radius at the rate of 1 revolution every 2 s. Find the ball's centripetal acceleration.

2

24

T

r

2

2

)2(

)6.0(4

s

m

= 5.92 m/s2

r

vac

2

CENTRIPETAL FORCE The inward force necessary to maintain uniform circular motion is defined as centripetal force. From Newton's Second Law, the centripetal force is given by:

Fmv

rc 2

Fc = macOr

Fc = m2

24

T

r Fc = m 4 π2 r f2

A 1000-kg car rounds a turn of radius 30 m at a velocity of 9 m/sa. How much centripetal force is required?

b. Where does this force come from?

Force of friction between tires and road.

m = 1000 kgr = 30 mv = 9 m/s

r

mvFc

2

m

smkg

30

)/9)(1000( 2

= 2700 N

Please complete PSE Practice Problems # 1-12Due EOC next class