Circular Motion

Circular Angular Measurement

0º

90º

180º

270º

(360º)

Degrees

π/2 rad

π rad

(3 π/2) rad

0 (2π rad)

pi=π=3.14159ratio of a circle’s circumferenceto the diameter.π=C/d radians is abbreviated rad.

What is a radian?• 1 radian – the angle contained in a distance along the

circumference of the circle (arc length) that is equal to the radius length.

r

s

57.3º

C=πdsince d=2rC=2πr

1 rad=57.3º

s=rθ, θ=angle in radians

1 rad = 360º/2π=180º/π

s=arc lengthr=radius of the circle

Conversion of Degrees to Radian and Radians to Degrees

• Radians x (180º/π)=Degrees

• Degrees x (π/180º)=Radians

Example:

1) 1.26 radians= ? degrees 1.26 rad (180º/π) = 72.2º

2) 254º = ? rad 254º (π/180º) = 4.43 rad

Relating the Arc Length, Radius, and Angle of a Circle

• s=rθ

What is the arc length based on an angle of 1.92 rad in a circle with a radius of 3.6 m?

s=(3.6 m)(1.92 rad) = 6.9m

s

3.6m

1.92 rad

Angular Position, Angular Distance, Angular Displacement and Linear Distance

r

s1

θ1

Angular Distance traveled until t1 from start: θ1

Linear Distance travel from start to t1: s=rθ d=rθ1

Linear Distance traveled between t1 and t2:S=d=s2-s1=rθ2-rθ1=r(θ2-θ1)

Angular Position at t1:θ1 (with respect to reference)

Angular Position at t2:θ2 (with respect to reference)

Angular Displacement between t1 and t2:Δθ=θ2-θ1

reference (0 rad)

θ2

t1t2

∆θ

s

Circular Position Equations

Angular Displacement between locations:

Δθ=θ2-θ1 (0 to 2π)

Linear Distance (arc length):

s=rθ=ds= arc length (linear distance)r=radiusθ = angular distance

Angular Position, Angular Distance, Angular Displacement and Linear Distance

100 m

1.9 rad

A person starts at a specific location on a circular track, travels once around the track and then ends at the location depicted in the diagram below. What are the angular position, distance, displacement and linear distance traveled?

Angular position:1.2 rad

Angular distance(1.9+2π) rad8.2 rad

Angular displacement:1.9 rad CCW

Linear distance traveled:s=rθ=(100m)8.2 rad=820 m

orC=2πr=2π(100)m =628 ms=rθ=100 m(1.9 rad)s=190 mdT=628 m+190m=8.2x102 m

s

reference (0 rad)

start

1.2 rad

Angular Speed, Velocity, and Tangential Velocity

ω=θ/tω = angular speed (measured in rad/s)θ = angular distance (rad)

12

12

ttt

locityangular ve Δθ = angular displacement (0 to 2π)

s=rθ s/t=r(θ/t) v=rω

v=rω

v = tangential velocity/speed (linear velocity/speed)

v

ω

r

Period, Frequency and Angular Velocity

T=period –the amount of time for one revolution or rotation.• period is measured in seconds.

ω=2π/T (based on one revolution)

f=frequency – the number of revolutions or rotations in one second.•frequency is measured in rev/s, rot/s, cycles/sec, s-1, or Hertz (Hz).

T=1/f f=1/T

ω=θ/t

ω=2πf

The Right Hand Rule

Curl the finger in the direction of rotation and note the direction of the thumb.

+ : Thumb points towards rotating object.- :Thumb points away from rotating object.

r1

r2

2

1

ω1=ω2

v2>v1

Angular and Tangential Velocity Relationship at Different Radii

Objects with the same angular speed revolving around the same central axis a have greater speed the farther away from the central axis.