Circular motion

A2

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

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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:

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Angular displacement can therefore be written in two ways:

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

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

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Therefore:

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

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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.

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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:

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From the diagram:

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

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And similarly for acceleration and angular acceleration:

circular motion 14

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SUVAT

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

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

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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:

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

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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:

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Torque

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

blank spacetorque = force x perpendicular distance

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

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We can now derive another equation for torque:

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Angular momentum

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

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

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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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