Question

Describe the direction, speed, and acceleration of an object moving with uniform circular motion.

Answer 1

For uniform circular motion (motion around a fixed circle at constant speed):

Direction of velocity
- The instantaneous velocity is always tangent to the circle.
- It is perpendicular to the radius drawn from the center to the object.
- If the object moves with angular speed ω and the angular position is θ(t)=ωt, v is in the θ̂ (azimuthal) direction.

Speed
- The speed is constant.
- Linear (tangential) speed v = ω r, where r is the circle radius.
- Equivalently v = 2πr / T = 2πr f, with period T or frequency f.

Acceleration
- There is a centripetal acceleration directed radially inward (toward the center of the circle).
- Its magnitude is a = v^2 / r = ω^2 r.
- The acceleration is perpendicular to the velocity, so it changes the direction of v but not its magnitude (hence constant speed).
- In vector form, if the position is r(t) = r(cos ωt, sin ωt), then v = rω(−sin ωt, cos ωt) and a = −rω^2(cos ωt, sin ωt) = −ω^2 r(t), showing a points toward the center.

Summary: velocity — tangent, constant magnitude v = ωr; acceleration — radially inward, magnitude v^2/r (or ω^2 r), perpendicular to velocity.