Circular Motion

Definition of Circular Motion

Circular motion occurs when an object moves along a circular path, always maintaining a constant distance (radius) from a fixed central point. The path is curved, not straight, so the direction of the object is continuously changing even if its speed remains constant.

Examples of circular motion include:

• Wheels rotating on a car or bicycle
• Planets orbiting the Sun
• A ball tied to a string being swung around

Centripetal Force

For an object to move in a circle, there must be a force acting towards the centre of the circle. This force is called the centripetal force. It constantly pulls the object inward, preventing it from moving off in a straight line due to inertia.

Without centripetal force, the object would move in a straight line tangent to the circle.

Examples of centripetal forces include:

• Tension in a string when swinging a ball on a string
• Gravity acting on planets orbiting the Sun
• Friction between car tyres and road when turning

Speed and Velocity in Circular Motion

In circular motion, the object can move at a constant speed, but its velocity changes because velocity includes direction as well as speed.

The velocity vector is always tangent to the circular path, pointing in the direction the object is moving at that instant.

Because the direction of velocity changes continuously, the object is accelerating even if its speed is constant. This acceleration is called centripetal acceleration and it always points towards the centre of the circle.

This inward acceleration is what changes the direction of the velocity, keeping the object moving in a circle.

For instance, a car moving at a steady 20 m/s around a roundabout has a constant speed but its velocity changes direction continuously, so it is accelerating towards the centre of the roundabout.

Calculations in Circular Motion

The centripetal acceleration $a_c$ of an object moving in a circle is given by the formula:

$$a_c = \frac{v^2}{r}$$

For example, if an object moves at 3 m/s around a circle of radius 1.5 m, the centripetal acceleration is $a_c = \frac{3^2}{1.5} = 6 \text{ m/s}^2$.

• $v$ is the speed of the object in metres per second (m/s)
• $r$ is the radius of the circular path in metres (m)
• $a_c$ is the acceleration towards the centre in metres per second squared (m/s²)

This formula shows that centripetal acceleration increases with the square of the speed and decreases as the radius increases.

The centripetal force $F_c$ needed to keep an object moving in a circle is related to this acceleration by Newton’s second law (see Forces & their Interactions topic for more on Newton’s laws):

$$F_c = m a_c = m \frac{v^2}{r}$$

• $m$ is the mass of the object in kilograms (kg)
• $F_c$ is the centripetal force in newtons (N)

Typical values for circular motion vary widely depending on the context, for example:

• A car turning a corner might have a radius of a few metres and speeds of 10-30 m/s
• Planets orbiting the Sun have radii of millions of kilometres and speeds of tens of km/s

Example: If a car travels around a roundabout of radius 10 m at a speed of 5 m/s, the centripetal acceleration is:

$$a_c = \frac{5^2}{10} = \frac{25}{10} = 2.5 \text{ m/s}^2$$