Revise Circular motion for Cambridge 9231 (3.3)

To describe motion in a circle, it's more convenient to use angles and angular velocity than Cartesian coordinates. The angular displacement, $\theta$, is the angle swept out from a reference point, measured in radians. The rate of change of this angle is the angular velocity, $\omega$.

$\omega = \frac{d\theta}{dt}$

The linear speed, $v$, of a particle is related to its angular velocity and the radius $r$ of the circle. The velocity vector is always tangent to the path. Even if the speed $v$ is constant, the velocity is not, because its direction is changing. This change in velocity implies acceleration.

$v = r\omega$

According to Newton's Second Law, an acceleration must be caused by a net force. The acceleration directed towards the centre of the circle, $a_r$, is caused by a net force also directed towards the centre. This net force is called the centripetal force, $F_c$. It's crucial to understand that this is not a new, fundamental force but the resultant of existing physical forces like tension, gravity, friction, or normal reaction.

$F_c = ma_r = m\frac{v^2}{r} = mr\omega^2$

A classic problem involves a particle moving in a vertical circle, attached to a string or a light rod, or moving on the inside/outside of a circular track. Here, both speed and the required centripetal force change throughout the motion due to the influence of gravity. The principle of conservation of energy is often essential for finding the speed at different points.

When a vehicle turns on a flat road, the centripetal force is provided by friction. To reduce reliance on friction and allow for higher speeds, tracks are often banked. By tilting the surface, a component of the normal reaction force can be directed horizontally towards the centre of the turn, providing the necessary centripetal force.