# Circular motion

A2

An object is in uniform circular motion when it rotates at a steady rate.

Angular displacement, θ (greek letter theta), is the angle traveled by an object in time, t. c is the circumference of the circle and r is the radius. is the period (the time for one complete circle) and f is the frequency. These are related by:

Angular displacement can therefore be written in two ways:

Angular speed, ω (greek letter omega), is the angular displacement per second.

If kinetic friction does no work, ie the object is rolling without slipping:

Therefore:

Speed is constant, but since the direction is changing the object is accelerating towards the centre of the circle.

Acceleration must be caused by a force, since F = ma. This is the centripetal force, which acts towards the centre of the circle (this is called a radial force). This is often provided by tension in a piece of string, friction between a road and tyres, or gravity.

when mass is constant.

Forces in circular motion can be determined using the diagrams shown in forces.

University

If the object is not rotating at constant speed there is also a tangential force (at right angles to the radial component), causing it to accelerate. This is called angular acceleration, α.

Some of the above equations also need to be derived:

From the diagram:

By differentiating, we get the relationship between speed and angular speed:

And similarly for acceleration and angular acceleration:

SUVAT

If angular acceleration is constant, angular versions of the SUVAT equations can be used.

(Notice these are the same as the original SUVAT equations, with the symbols replaced:

Any values with a subscript 0 mean that this is the initial, eg θ0 is the initial angular displacement.

Energy

Rotating objects have rotational kinetic energy, given by:

Here, I is analogous to m. I is called the moment of inertia. This is calculated using:

This is easy with a small number of particles, and gets trickier with more, so that integration is needed.

Total kinetic energy of a rotating object is the sum of the rotational and translational kinetic energies:

Torque

A torque, τ (greek letter tau), is a turning force:

torque = force x perpendicular distance

This is given by a cross product, since the distance and force must be at right angles.

We can now derive another equation for torque:

Angular momentum

Angular momentum, L, is analogous to linear momentum, p:

If we differentiate with respect to time, we end up with torque:

This means that angular momentum is conserved when there is zero torque (angular momentum remains constant).

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