Motion in a Circle

We shall look at two different cases of motion in a circle: one where the velocity is of constant magnitude (uniform motion) and one where the magnitude of $\vec v$ varies (non-uniform motion).

Uniform Circular Motion

Please use this applet for a demonstration of uniform circular motion before reading the rest of this page.

Uniform circular motion means that an object (or particle) moves in a circle with constant speed, e.g., a car going round a roundabout at constant speed, or a satellite orbiting the earth. There is no component of acceleration parallel to the path, i.e. tangent to the circle, otherwise the speed would increase: In uniform circular motion, the acceleration is perpendicular to the velocity at every instant and is directed towards the centre of the circle.

The particle moves from $P_1$ to $P_2$ in a time $\Delta t$. The change in the velocity is given by $$\Delta \vec v=\vec v_2-\vec v_1\,.$$ We can now simplify the calculation by moving the velocity vectors so that they share a starting point:

Note that the angles labelled $\Delta\varphi$ in the two figures above are indeed the same, because $\vec v_1$ is perpendicular to the line $OP_1$ and $\vec v_2$ is perpendicular to the line $OP_2$, and thus the triangles $OP_1P_2$ and $Op_1p_2$ are similar. We also know that for similar triangles the ratio of corresponding sides are equal, so $$\frac{|\Delta \vec v|}{v_1}=\frac{\Delta s}{R}=\Delta\varphi\,;$$ (the final equality only holds for small $\Delta\varphi$). Thus the magnitude of the average acceleration is $$a_{\text{av}}=\frac{|\Delta \vec v|}{\Delta t}=\frac{v_1}{R}\frac{\Delta s}{\Delta t}\,.$$

In the limit that $P_2$ approaches $P_1$ and $\Delta t \rightarrow 0$ we find that the magnitude of the instantaneous acceleration becomes $$a=\lim_{\Delta t\rightarrow 0} \frac{v_1}{R}\frac{\Delta s}{\Delta t}=\frac{v}{R}\frac{\dd s}{\dd t}\,.$$ In the last term we have dropped the subscript "$1$" on $v$ since $P_1$ approaches $P_2$, and we might consider this the generic point $P$. We thus conclude that $$a=v^2/R\,.$$

For uniform circular motion the vector $\vec a$ is always directed along the radius towards centre of circle.

Non-Uniform Circular Motion

We speak of non-uniform motion when the speed varies, i.e., it is not constant. Even in this case, the acceleration has a radial component linked to velocity: $$a_{\text{rad}}=\frac{v^2}{R}\,,$$ which is by definition perpendicular to the velocity.

But $a_{\text{rad}}$ is no longer constant – it varies as the speed, $v$, varies – the bigger $v$, the bigger the acceleration. We conclude that there is now also a component of $\vec a$ which is parallel to the motion, i.e. tangent to the circle. From our previous discussion, the parallel or tangential component of $\vec a$ is equal to the rate of change of speed, i.e. $$a_{\text{tan}}=\frac{\dd|\vec v|}{\dd t}\,,$$

The vector $\vec a$ is equal to the vector sum of ${\vec a}_{\text{rad}}$ and ${\vec a}_{\text{tan}}$; ${\vec a}_{\text{tan}}$ is in same direction as the velocity if the particle is speeding up, and in the opposite direction to the velocity if particle is slowing down.

Click here for an interative demonstration of the motion of a particle in a vertical loop.