Revise Centripetal acceleration for Cambridge 9702 (12.2)

Imagine an object tracing a perfect circle. While its speedometer might read a steady value, its motion isn't uniform in the purest sense. Velocity, being a vector, is defined by both how fast an object is moving and in which direction. In circular motion, the direction of travel is continuously shifting. This continuous alteration of the velocity vector means there's a constant rate of change of velocity, which is, by definition, an acceleration.

This unique acceleration experienced by objects in circular motion is termed centripetal acceleration ($a_c$). Crucially, its direction is always perpendicular to the instantaneous linear velocity of the object and points directly towards the absolute centre of the circular path. Without this inward acceleration, the object would simply fly off tangentially in a straight line, as dictated by Newton's First Law of Motion.

To quantify centripetal acceleration, we can use the object's linear speed. Linear speed ($v$) is how fast the object is moving along the circumference, measured in metres per second (m s⁻¹). The radius ($r$) is the distance from the object to the centre of its circular path, measured in metres (m).

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

Alternatively, centripetal acceleration can be expressed using angular speed. Angular speed ($\omega$) is the rate at which the object's angular position changes, measured in radians per second (rad s⁻¹). This formula is particularly useful when dealing with rotational systems.

$a_c = \omega^2 r$

According to Newton's Second Law ($F=ma$), if there's an acceleration, there must be a resultant force causing it. This resultant force, which is responsible for maintaining circular motion, is known as the centripetal force ($F_c$). Just like centripetal acceleration, the centripetal force always acts towards the centre of the circular path. It’s important to remember that centripetal force isn't a new fundamental force; rather, it’s a role played by existing forces like tension, gravity, or friction.

$F_c = m a_c$

By substituting the expressions for $a_c$, we get two primary formulas for centripetal force:

$F_c = \frac{mv^2}{r}$

$F_c = m\omega^2 r$