Centripetal Force (Edexcel A Level Physics): Revision Note

Centripetal Force

•  Centripetal force can be calculated using any of the following equations:

$F=\frac{m{v}^{\mathit{2}}}{r}=mr{\omega }^2=mv\omega$

• Where:

  • F = centripetal force (N)

  • v = linear velocity (m s^-1)

  • ⍵ = angular speed (rad s^-1)

  • r = radius of the orbit (m)

Centripetal force is always perpendicular to the direction of travel

• The centripetal force is the resultant force on the object moving in a circle

  • This is particularly important if there are multiple forces on the object, such as weight

Vertical Circular Motion

• An example of vertical circular motion is swinging a ball on a string in a vertical circle

• The forces acting on the ball are:

  • The tension in the string

  • The weight of the ball, acting downwards

• As the ball moves around the circle, the direction of the tension changes continuously, while the weight always acts downwards

• By conservation of mechanical energy, the ball is:

  • fastest at the bottom (where its gravitational potential energy is lowest)

  • slowest at the top (where its potential energy is highest)

• The magnitude of the tension, therefore, varies, reaching a maximum at the bottom and a minimum at the top

• At the bottom of the circle:

  • The centripetal force is directed upwards (towards the centre of the circle)

  • The tension acts upwards, and the weight acts downwards

• So, the tension must support the weight and provide the centripetal force:

$T_{max}-mg=\frac{m{v}_{bottom}^2}{r}$

$T_{max}=\frac{m{v}_{bottom}^2}{r}+mg$

• At the top of the circle:

  • The centripetal force is directed downwards (towards the centre of the circle)

  • The tension and the weight both act downwards

• So, together the tension and the weight provide the centripetal force:

$T_{min}+mg=\frac{m{v}_{top}^2}{r}$

$T_{min}=\frac{m{v}_{top}^2}{r}-mg$

• The tension is therefore greatest at the bottom for two reasons:

  • $v^2$ is largest at the bottom (from conservation of energy)

  • At the bottom, weight is subtracted from the centripetal force requirement to give the tension; at the top, weight is added — so for the same speed, T at the bottom would already exceed T at the top