Circular Motion

Circular Angular Measurement 0º0º 90º 180º 270º (360º) Degrees π/2 rad π rad (3 π /2) rad 0 (2π rad) pi=π= ratio of a circle’s circumference to 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=πd since d=2r C=2πr 1 rad=57.3º s=rθ, θ=angle in radians 1 rad = 360º/2π=180º/π s=arc length r=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 s1s1 θ1θ1 Angular Distance traveled until t 1 from start: θ 1 Linear Distance travel from start to t 1 : s=rθ  d=rθ 1 Linear Distance traveled between t 1 and t 2 : S=d=s 2 -s 1 =rθ 2 -rθ 1 =r(θ 2 -θ 1 ) Angular Position at t 1 : θ 1 (with respect to reference) Angular Position at t 2 : θ 2 (with respect to reference) Angular Displacement between t 1 and t 2 : Δθ=θ 2 -θ 1 reference (0 rad) θ2θ2 t1t1 t2t2 ∆θ∆θ s

Circular Position Equations Angular Displacement between locations: Δθ=θ 2 -θ 1 (0 to 2π) Linear Distance (arc length): s=rθ=d s= arc length (linear distance) r=radius θ = 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) rad 8.2 rad Angular displacement: 1.9 rad CCW Linear distance traveled: s=rθ=(100m)8.2 rad=820 m or C=2πr=2π(100)m =628 m s=rθ=100 m(1.9 rad) s=190 m d T =628 m+190m=8.2x10 2 m s reference (0 rad) start 1.2 rad

Angular Speed, Velocity, and Tangential Velocity ω=θ/t ω = angular speed (measured in rad/s) θ = angular distance (rad) 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.

r1r1 r2r2 2 1 ω 1 =ω 2 v 2 >v 1 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.