3.3 Circular motion

Angular speed ω measures the angle swept per unit time, in radians per second. For a particle at radius r, its tangential speed is v=rω, so equal angular speed does not mean equal linear speed at different radii.

Use radians, not degrees, in v=rω and in arc-length relations. Keep the direction of v tangent to the circle even though ω describes the rotation about the centre.

At ω=4 rad s⁻¹, a point 0.25 m from the axis moves at v=1.0 m s⁻¹; a point twice as far moves twice as fast.

Angular speed is not the same as revolutions per second: f revolutions per second gives ω=2πf.

For uniform circular motion the velocity direction continually changes, so acceleration is directed towards the centre. Its magnitude is a=v²/r=rω².

The inward acceleration is supplied by the resultant inward force, not by a new separate force. Draw the radial direction first, then apply Newton’s second law along it.

A 0.50 kg mass moving at 3.0 m s⁻¹ on a 2.0 m radius circle needs a=4.5 m s⁻² inward and resultant force 2.25 N.

Constant speed does not mean zero acceleration; only the magnitude of velocity is constant, while its direction changes.

For a particle moving at constant speed in a horizontal circle, the resultant horizontal or radial force must equal mv²/r. Vertical forces must separately balance if the height is constant.

Resolve the actual forces—tension, normal reaction, friction or a component of weight—before setting their radial resultant equal to mv²/r. The centre direction changes around the circle.

For a conical pendulum, the vertical component of tension balances mg while the horizontal component supplies m v²/r; using all of T as centripetal force is wrong.

“Centripetal force” is a role played by the resultant inward force, not an extra force to add to the free-body diagram.

In a vertical circle, speed changes with height. Use conservation of energy between points, then apply radial Newton’s law to find tension or normal reaction at that point.

At the top and bottom, define inward separately: tension or reaction may add to weight at the bottom but oppose it at the top. Complete contact requires the limiting reaction to remain non-negative.

For a particle on a string, the minimum speed at the top occurs when tension is zero, giving mv²/r=mg there; energy then determines the required bottom speed.

The condition T=0 is a limiting contact condition, not “no gravity”, and using one fixed speed around the circle violates energy conservation.