Centripetal Force

Copyright © David Boettcher 2005 - 2023 all rights reserved.

If a mass is forced to follow a circular path, such as the mass \(m\) in figure 1, it is continually deflected from the straight path tangential to the circle, the path that its momentum would otherwise carry it along.

Velocity is not the same as speed; it is a vector rather than scalar measurement that includes speed and direction. The momentum of the mass is constantly changing, because the direction component of its velocity is constantly changing even if its speed remains the same.

According to Newton's laws of motion, a change in momentum must be caused by a force. Newton said that the force is equal to the rate at which the object’s “progress” (today called momentum) changes, which is why momentum is abbreviated as “p”.

In the case of the rotating mass shown in figure 1, the force \(F_c\) keeps the mass at a constant radius from the centre of the circle of rotation, constantly changing its direction, and is called centripetal force.

The velocity at points 1 and 2 is \(v_1\) and \(v_2\). The difference between these velocities \(\Delta v \) can be found by the difference between their magnitudes and directions as shown in the small vector diagram. Because both \(v_1\) and \(v_2\) are tangential to a radius, the angle between their vectors \( \theta \) is the same as the angle \(\theta\) subtended by points 1 and 2 at the centre of rotation. The two triangles are therefore similar, and the ratio of the length of their bases to their sides is equal.

If the time interval \(\Delta t\) between points 1 and 2 is small, the distance between points 1 and 2 on the circle can be approximated as \(v\Delta t\).

The lengths of \(v_1\) and \(v_2\) are the same \(v\). The ratio of \(\Delta v \) to \(v\) is the same as the ratio of the distance on the circle between points 1 and 2, \(v\Delta t\), to \(r\). That is;

\begin{equation} \frac{\Delta v}{v}=\frac{v\Delta t}{r} \end{equation}

From Newton's laws of motion, the centripetal acceleration \(a_c\) of the mass \(m\) is the rate of change of its velocity; \( \frac{\Delta v}{\Delta t} \).

\begin{equation} a_c = \frac{\Delta v}{\Delta t}=\frac{v^2}{r} \end{equation}

From Newton's Second Law, \(F=ma\), the centripetal force \(F_c\) is:

\[ F_c = m a_c = \frac{mv^2}{r} \]

The centripetal force is developed in whatever it is, a piece of rope, a steel bar, etc., that is causing the mass to travel in the arc of a circle.

It is often said that rotation causes a centrifugal force that tries to make the mass fly outwards, implying that the mass would move radially if this force was not opposed. But in reality, if the centripetal force suddenly ceased to exist, the mass would not fly off along a radius to the circle of rotation but would instead continue in a straight line at a tangent to the circle. Centripetal force causes the mass to deviate from this tangential path.

However, it is often easier to think in terms of a centrifugal force, even if it is “imaginary”.