Transcript Slide 1
Circular Motion Circular Angular Measurement Degrees 90º π/2 rad 0º (360º) 180º π rad 0 (2π rad) 270º (3 π/2) rad pi=π=3.14159 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. 1 rad=57.3º 57.3º C=πd since d=2r C=2πr r 1 rad = 360º/2π=180º/π s s=arc length r=radius of the circle s=rθ, θ=angle in radians 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 • s=rθ 1.92 rad What is the arc length based on an angle of 1.92 rad in a circle with a radius of 3.6 m? 3.6m s=(3.6 m)(1.92 rad) = 6.9m Angular Position, Angular Distance, Angular Displacement and Linear Distance s Angular Position at t1: θ1 (with respect to reference) t1 t2 Angular Position at t2: θ2 (with respect to reference) s1 θ2 ∆θ θ1 reference (0 rad) r Angular Displacement between t1 and t2: Δθ=θ2-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) Circular Position Equations Angular Displacement between locations: Δθ=θ2-θ1 (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 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? s 100 m 1.2 rad Angular position: 1.2 rad Angular distance (1.9+2π) rad 8.2 rad Angular displacement: 1.9 rad CCW 1.9 rad reference (0 rad) start 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 dT=628 m+190m=8.2x102 m Angular Speed, Velocity, and Tangential Velocity ω=θ/t ω = angular speed (measured in rad/s) θ = angular distance (rad) 2 1 t t 2 t1 angular ve locity 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 ω=θ/t 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=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. Angular and Tangential Velocity Relationship at Different Radii 2 1 r2 r1 ω1=ω2 v2>v1 Objects with the same angular speed revolving around the same central axis a have greater speed the farther away from the central axis.