__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. *T *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 θ_{0} 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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