Edexcel IAL - Mechanics 3- 4.3 Uniform Circular Motion- Study notes  - New syllabus

Uniform Motion in a Horizontal Circle

A particle moves with uniform circular motion if it travels at constant speed in a horizontal circular path. Although the speed is constant, the velocity is continually changing in direction, so the particle has an acceleration towards the centre of the circle.

This inward acceleration is the radial (centripetal) acceleration.

Radial Acceleration and Force

For a particle of mass \( m \) moving in a horizontal circle of radius \( r \) with speed \( v \) and angular speed \( \omega \):

Radial acceleration: \( a = \dfrac{v^2}{r} = r\omega^2 \)

Radial force: \( F = \dfrac{mv^2}{r} = mr\omega^2 \)

The resultant horizontal force acting towards the centre must provide this radial force.

Conical Pendulum

A conical pendulum consists of a particle attached to a light inextensible string of length \( l \), moving in a horizontal circle so that the string makes a constant angle \( \theta \) with the vertical.

Forces acting on the particle:

Weight \( mg \) acting vertically downward

Tension \( T \) in the string acting along the string

Resolving forces:

Vertical: \( T\cos\theta = mg \)

Horizontal: \( T\sin\theta = \dfrac{mv^2}{r} \)

Elastic String in Circular Motion

If a particle moves in a horizontal circle attached to an elastic string:

• The tension in the string provides the centripetal force
• The tension depends on the extension of the string

If the natural length is \( l \), the stretched length is \( x \), and the modulus of elasticity is \( \lambda \):

\( T = \dfrac{\lambda}{l}(x – l) \)

Motion on a Banked Surface

When a particle moves in a horizontal circle on a smooth banked surface inclined at angle \( \theta \) to the horizontal:

The normal reaction provides the centripetal force

Resolving the normal reaction \( N \):

Vertical: \( N\cos\theta = mg \)

Horizontal: \( N\sin\theta = \dfrac{mv^2}{r} \)