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Circular Motion

Circular Motion. Degrees. Circular Angular Measurement. 90 º. π /2 rad. 0 º. (360 º ). 180 º. 0 (2 π rad). π rad. pi= π =3.14159 ratio of a circle’s circumference to the diameter. π =C/d radians is abbreviated rad. 270 º. (3 π /2) rad. What is a radian?.

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Circular Motion

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  1. Circular Motion

  2. Degrees Circular Angular Measurement 90º π/2 rad 0º (360º) 180º 0 (2π rad) π rad pi=π=3.14159 ratio of a circle’s circumference to the diameter. π=C/d radians is abbreviated rad. 270º (3 π/2) rad

  3. 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 C=πd since d=2r C=2πr s 57.3º r 1 rad = 360º/2π=180º/π s=rθ, θ=angle in radians s=arc length r=radius of the circle

  4. Conversion of Degrees to Radian and Radians to Degrees • Radians x (180º/π)=Degrees • Degrees x (π/180º)=Radians • Example: • 1.26 radians= ? degrees • 1.26 rad (180º/π) = 72.2º • 2) 254º = ? rad • 254º (π/180º) = 4.43 rad

  5. Relating the Arc Length, Radius, and Angle of a Circle • s=rθ s 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

  6. Angular Position, Angular Distance, Angular Displacement and Linear Distance s 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 t1 t2 s1 θ2 ∆θ θ1 reference (0 rad) r 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)

  7. 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

  8. 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? 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 dT=628 m+190m=8.2x102 m s 100 m 1.2 rad 1.9 rad reference (0 rad) start

  9. Angular Speed, Velocity, and Tangential Velocity ω=θ/t ω = angular speed (measured in rad/s) θ = angular distance (rad) v Δθ = angular displacement (0 to 2π) ω r s=rθ s/t=r(θ/t)  v=rω v=rω v = tangential velocity/speed (linear velocity/speed)

  10. 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

  11. 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.

  12. 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.

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