+ Circular Motion

+Circular Motion

!  Uniform circular motion (constant centripetal acceleration)

!  Motion with a tangential and radial (centripetal)acceleration

+Uniform Circular Motion

! Object has a circular path, constant speed.

! The velocity is always tangent to the path of the object.

Why is there acceleration in uniform circular motion?

Δv = vf – vi

a= Δv/ Δt

+Centripetal Acceleration !  Vector

!  Always perpendicular to the path of the motion.

!  Points toward the center of the circle.

2

Cvar

=

+Characteristics of UniformCircular Motion

!  Tangential (linear) Velocity

!  Frequency

!  Period

!  Centripetal Acceleration

+ Frequency, f : #revolutions per unit time

!  f = # rev / time

Units:

!  (1/sec)=sec-1=Hertz (Hz)

!  rpm (#rev/min)

!  rps (#rev/sec)

r

+Period

!  Period T : time for 1 revolution !  Unit: sec, min, h

!  Relating Frequency and period f= 1

T

+Tangential Speed in terms of T or f

v=2πr/T

v=2πrf

+Proof of a = v2/r

+Example 1

! A boy whirls a stone in a horizontal circle of r=1.5m, 2m above the ground. The string breaks and the stone strikes 10m away. What was the centripetal acceleration during the circular motion?

!  Answer: 160m/s2

+ When Acceleration is not uniform -Tangential Acceleration exists

! When v varies.

! Motion is not uniform circular motion.

Total Acceleration

! Tangential acceleration (vector)

Plus

! Radial acceleration (vector)

+Total Acceleration

!  Tangential acceleration:

!  Radial acceleration:

!  Total acceleration:

td

adt

=v

2

r Cva ar

= − = −

2 2r ta a a= +

Total Acceleration

! r, radius vector - θ , tangent to the circle

2ˆ ˆt r

d vdt r

θ= + = −v

a a a r

+Example 2

An automobile whose speed is

increasing at a rate of 0.6m/s2 travels on

a circular road of radius 20m. When the

instantaneous speed of the car is 4m/s

find: a)the tangential acceleration, b)the

centripetal acceleration, c)the magnitude

and direction of the total acceleration.

Ans: a)0.6m/s2,b)0.8m/s2, c)1m/s2,53.1o