Circular Motion Basics

When a particle moves in a circle, it is constantly changing direction, which means it is accelerating even if its speed is constant. This centripetal acceleration is directed towards the centre of the circle and requires a centripetal force.

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1 Angular Speed

If a particle moves in a circle of radius r, its angular speed omega is:

omega = d theta/dt

where theta is the angle swept out. The SI unit is rad s^(-1).

For one complete revolution: theta = 2pi, time = T (period).

omega = 2pi/T = 2pi f

where f is the frequency (revolutions per second).

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2 Relationship Between Linear and Angular Speed

The linear speed v (tangential speed) is related to angular speed by:

v = r omega

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3 Centripetal Acceleration

A particle moving at constant speed v in a circle of radius r has acceleration directed towards the centre:

a = v^2/r = r omega^2

This is the centripetal acceleration. It changes the direction of the velocity but not its magnitude.

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4 Centripetal Force

By Newton's Second Law, the force required to maintain circular motion is:

F = mv^2/r = mr omega^2

This is not a new type of force -- it is the resultant of all forces acting on the particle, directed towards the centre. The centripetal force might be provided by tension, gravity, friction, normal reaction, or a combination.

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5 Worked Example 1

A particle of mass 0.5 kg moves in a horizontal circle of radius 2 m at 3 rad s^(-1). Find the centripetal force.

Solution:

F = mr omega^2 = 0.5 * 2 * 9 = 9 N

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6 Worked Example 2

A car travels around a circular bend of radius 50 m at 20 m s^(-1). Find the centripetal acceleration.

Solution:

a = v^2/r = 400/50 = 8 m s^(-2)

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7 Worked Example 3

A satellite orbits Earth at height h above the surface. The gravitational force provides the centripetal force. Show that v = sqrt(gR^2/(R + h)) where R is Earth's radius.

Solution:

Gravitational force at height h: F = GMm/(R+h)^2 = mg R^2/(R+h)^2

Centripetal force: F = mv^2/(R+h)

Setting equal: mg R^2/(R+h)^2 = mv^2/(R+h)

v^2 = gR^2/(R+h)

v = sqrt(gR^2/(R + h))

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8 Period and Frequency

Period T = time for one revolution = 2pi/omega = 2pi r/v