Circular motion


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

circular motion 9

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:

circular motion 1

Angular displacement can therefore be written in two ways:

circular motion 2

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

circular motion 3

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

circular motion 4


circular motion 5

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

circular motion 6

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.

circular motion 7

when mass is constant.

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


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:

circular motion 10

From the diagram:

circular motion 11

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

circular motion 12

And similarly for acceleration and angular acceleration:

circular motion 14

circular motion 13


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

circular motion 15

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

circular motion 24.jpg

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


Rotating objects have rotational kinetic energy, given by:

circular motion 16

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

circular motion 18

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:

circular motion 17


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

blank spacetorque = force x perpendicular distance

circular motion 19

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

circular motion 20

We can now derive another equation for torque:

circular motion 21

Angular momentum

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

circular motion 22

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

circular motion 23

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

Back to Contents: Physics: Mechanics

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