Radial Acceleration and Centripetal Force

Objects travelling in a Circle

Newton's First Law states :-

"An object will remain at rest or travel in a straight line with a constant speed, unless an unbalanced Force acts upon it"

This implies that any object that moves in a circle therefore must be experiencing an unbalanced Force, as shown in the diagram below :-

Note - As an unbalanced Force is acting on the object, the object is continuously accelerating. As the Force is at a right angle to the tangential motion of the object, the magnitude of the Velocity is constant but the direction is changing. This means an object can accelerate without changing speed.

This acceleration toward the centre of the circle is known as the Radial Acceleration and requires an inwards Force to cause it - the Centripetal Force.


Derivation of Radial Acceleration

In the above diagram, a particle moves along the Arc AB in a time of Δt, with a velocity of v.

The time ( Δt ) can be found using the following formula :-

The Change in Velocity ( Δv ) can be found by the following approximation :-

The Average Acceleration of the particle can then be found by the following :-

The above formula gives the Acceleration over a large time period. If the value of θ is made to tend towards Zero, the value of sinθ ≈ θ and the Instantaneous Acceleration at a point can be found :-

Note - The direction of this Acceleration is always towards the centre of the Circular Motion.


Centripetal Force

By following Newton's Second Law, in order for an object to experience an Acceleration, a Force must be applied to it. In the case of Radial Acceleration, that Force is the Centripetal Force. The Centripetal Force can be applied through several different ways :-

Frictional Force provided by tyres to keep car travelling in a circular path

Gravitational Force keeping a Satellite travelling in a circular orbit

Electrostatic Force keeping an Electron in a circular orbit ( Classical Mechanics )

Reaction Force from the wall keeping a Wall of Death rider in a circular path

The Centripetal Force can be found mathematically by combining Newton's Second law with the Radial Acceleration :-

Centripetal Force Case Study : Horizontal Circle

A mass can be attached to the end of a string and rotated in a horizontal circle as shown in the diagram below :-

If the mass is made to rotate around in a horizontal circle, the Centripetal Force required to keep the mass in the circular path is provided by the Tension in the string, which is in turn provided by the Weight of the masses on the hanger. If the mass is made to rotate at a rate at which the string does not move vertically through the glass tube, the above relationship for the Centripetal Force can be confirmed by measuring the rate of rotation and finding the Angular Velocity.

Note - When this experiment is actually performed, due to the weight acting on the mass, the mass is not moving in a circle of radius r, but one slightly smaller. The path actually taken can be seen in the Conical Pendulum case study below.


Centripetal Force Case Study : Vertical Circle

A mass can be attached to the end of a string and rotated in a vertical circle as shown in the diagram below :-

If the mass is made to rotate around in a vertical circle, the Centripetal Force required to keep the mass in the circular path is again provided by the Tension in the string. The Tension in the string is not constant in this motion. This is due to Weight acting either with or against the Tension at different points in the rotation :-

Position 1 - Centripetal Force = TMinimum + Weight

Position 2 - Centripetal Force = T2

Position 3 - Centripetal Force = TMaximum - Weight

Note - If the object is travelling in a circle, the magnitude of the Centripetal Force must be constant, therefore the Tension must change throughout the rotation. This means the string is most likely to snap at the bottom of the rotation, when the Tension in the string is at maximum.


Centripetal Force Case Study : Conical Pendulum

As was seen in the horizontal circle case study above, Weight caused the object to follow the path shown in blue below :-

As can be seen from the diagram above the Tension acts at an angle to the circular motion. This means that it is a component of the Tension ( Tsinθ ) which acts as the Centripetal Force in this case.


Centripetal Force Case Study : Car on a Circular Track

For a car moving on a circular track ( or around a roundabout ), the Centripetal Force is provided by the Frictional Force between tyres and the road.

As the Velocity increases, the Centripetal Force required to keep the car travelling in a circle increases. When the Frictional Forces can no longer provide the required Centripetal Force, the car skids and follow a tangential path off the Track.


Centripetal Force Case Study : Car on a Banked Circular Track

As can be seen in the above case study, there is a maximum tangential Velocity an object can have in order to remain in a circular path if the Centripetal Force is provided by Friction.

By changing the angle of the track, however, it is possible to achieve a higher speed than would be possible on the flat track. This is due to a component of the Reaction Force acting Centripetally :-

The above diagram shows a car on a banked track. The Weight acts vertically downwards with a magnitude of mg. The Reaction force provided by the track is shown as R and is at a right angle to the track.

In order to understand the effect of this Reaction Force acting at an angle, the components of the Reaction Force must be found.

As shown in the diagram above, the horizontal component of the Reaction Force acts towards the centre of the circle. This horizontal component increases as the angle of the ramp increases.

This means the steeper the ramp, the higher the maximum tangential Velocity can be.

Note - The above statement only applies until the car slides down the ramp or until the centre of Mass is beyond a critical pivot point, beyond which the car would topple over.

The video below shows an interview with the Chief Architect of the Olympic Velodrome in London discussing the Science behind the banked track :-