Revise Kinematics of uniform circular motion for Cambridge 9702 (12.1)

Uniform circular motion describes an object tracing a circular path while maintaining a constant speed. Crucially, even though its speed doesn't change, its direction is continuously changing. Because velocity is a vector (it has both magnitude and direction), a change in direction means the object's velocity is constantly changing. This continuous change in velocity implies that the object must be accelerating.

When an object moves in a circle, we can describe its position using the angle it has swept out from the center. This is called angular displacement ($\theta$). Instead of degrees, physicists often use a unit called the radian (rad) for angular measurements in circular motion.

$1\text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$

These terms help us describe how fast an object completes its circular path. The period ($T$) is simply the time taken for one complete revolution around the circle, measured in seconds. The frequency ($f$) is the number of revolutions completed per unit time, usually per second. They are intimately linked.

$f = \frac{1}{T}$

Just as linear velocity measures the rate of change of linear displacement, angular velocity ($\omega$) measures the rate of change of angular displacement. Its standard unit is radians per second (rad s⁻¹). Its direction is typically described as clockwise or anticlockwise.

$\omega = \frac{\Delta\theta}{\Delta t} = \frac{2\pi}{T} = 2\pi f$

The linear speed ($v$) of an object moving in a circle, which is tangent to the path, is directly related to its angular velocity ($\omega$) and the radius ($r$) of the circular path. A larger radius or faster angular spin results in a greater linear speed.

$v = \omega r$

This is a key concept! Even if an object is moving at a constant speed in a circle, its velocity is continuously changing because its direction is constantly turning. Since acceleration is defined as the rate of change of velocity, an object in uniform circular motion is always accelerating.

The acceleration an object experiences in uniform circular motion is called centripetal acceleration ($a_c$). The word 'centripetal' means 'centre-seeking'. This acceleration is always directed towards the exact center of the circular path, acting perpendicular to the object's instantaneous linear velocity.

$a_c = \frac{v^2}{r} = \omega^2 r$

According to Newton's Second Law, if an object is accelerating, there must be a resultant force acting on it. This force, which is necessary to maintain circular motion, is called centripetal force ($F_c$). Like centripetal acceleration, it is always directed towards the center of the circle, perpendicular to the linear velocity.

$F_c = ma_c = \frac{mv^2}{r} = m\omega^2 r$

It's crucial to understand that centripetal force is not a new, fundamental force of nature. It is the net force that points towards the center of the circular path. This net force is provided by one or more familiar forces.