Uniform Circular Motion

Position on a Circle Motion in a circle is common. The most important measure is the radius (r). The position of a point on the circle is described by a radial vector . Origin is at the center. Magnitude is equal everywhere. r r

Measuring a Circle We use degrees to measure position around the circle. There are 2p radians in the circle. This matches 360° The distance around a circle is s = r q, where q is in radians. Dq q r The angular displacement is Dq

Period and Frequency Movement around a circle takes time. The period (T) is the time it takes to complete one revolution around the circle. The frequency (f) is the number of cycles around completed in a time. Cycles per second (cps or Hz) Revolutions per minute (rpm) Frequency is the inverse of period (f = 1/T).

Cycles or Radians Frequency is measured in cycles per second. There is one cycle per period. Frequency is the inverse of the period, f =1/T. Angular velocity is measured in radians per second. There are 2p radians per period. Angular velocity, w = 2p/T. Angular velocity, w = 2pf.

Angular Velocity Displacement is related to the angle. Displacement on the curve (s) Angle around the circle (q) Velocity has an angular equivalent. Linear velocity (v) Angular velocity (w) Units (rad/s or 1/s = s-1)

Speed on a Circle The circumference of a circle is 2r. The period is T. The speed is related to the distance and the period or frequency. v = 2r/T v = 2rf v = rw s = 2p r r

Velocity on a Circle Velocity is a vector change in position compared to time. As the time gets shorter, the velocity gets closer to the tangent.

Direction of Motion In the limit of very small angular changes the velocity vector points along a tangent of the circle. This is perpendicular to the position. For constant w, the magnitude stays the same, but the direction always changes.

No Slipping A wheel can slide, but true rolling occurs without slipping. As it moves through one rotation it moves forward 2pR. w v v = 2pR/T = wR R Dx = 2pR

Point on the Edge A point on the edge moves with a speed compared to the center, v = wr. Rolling motion applies the same formula to the center of mass velocity, v = wR. The total velocity of points varies by position. v = 2vCM vCM v = 0