5.4.4 Centripetal Force

• For an object moving in a circle, it will have the following properties:
  • Period
  • Frequency
  • Angular displacement
  • Angular velocity

• These properties can be inferred from the properties of objects moving in a straight line combined with the geometry of a circle

Motion in a Straight Line

• When an object moves in a straight line at a constant speed its motion can be described as follows:
  • The object moves at a constant velocity, v
  • Constant velocity means zero acceleration, a
  • Newton's First Law of motion says the object will continue to travel in a straight line at a constant speed unless acted on by another force
  • Newton's Second Law of motion says for zero acceleration that there is no net or resultant force

• For example, an ice hockey puck moving across a flat frictionless ice rink

Motion in a Circle

• If one end of a string was attached to the puck, and the other attached to a fixed point, it would no longer travel in a straight line, it would begin to travel in a circle
• The motion of the puck can now be described as follows:
  • As the puck moves it stretches the string a little to a length r
  • The stretched string applies a force to the puck pulling it so that it moves in a circle of radius r around the fixed point

• The force acts at 90° to the velocity so there is no force component in the direction of velocity
  • As a result, the magnitude of the velocity is constant
  • However, the direction of the velocity changes
  • This means there is acceleration present in the circular motion, so there must be a net force 

• As it starts to move in a circle the tension of the string:
  • Continues to pull the puck at 90° to the linear velocity
  • Acts towards the centre of the circle
  • Is the only force acting on the puck
  • Hence, the net or overall force is towards the centre of the circle 

• So, the net force acting on the puck is called the centripetal force

Centripetal force and acceleration are always directed towards the centre of the circle

• The centripetal force is not a separate force of its own
  • It can be any type of force, depending on the situation, which keeps an object moving in a circular path

Examples of centripetal force

• The centripetal force, F, is defined as:

The resultant force towards the centre of the circle required to keep a body in uniform circular motion. It is always directed towards the centre of the body's rotation.

•  Centripetal force can be calculated using:

Centripetal force is always perpendicular to the direction of travel

• Where:
  • F = centripetal force (N)
  • v = linear velocity (m s^-1)
  • ⍵ = angular speed (rad s^-1)
  • r = radius of the orbit (m)

Step 1: Draw the forces on the bucket at the top

Step 2: Calculate the centripetal force

• • The weight of the bucket = mg
  • This is equal to the centripetal force since it is directed towards the centre of the circle

Step 3: Rearrange for velocity v

• • m cancels from both sides

Step 4: Substitute in values