Characteristics of Circular Motion

Characteristics of Circular Motion

Two characteristics are stated at the very start of the section, and both are examinable one-liners.

• It is an accelerated motion. The direction of the velocity changes at every instant, so even at constant speed the velocity vector changes and the motion is accelerated.
• It is a periodic motion. The particle repeats its path along the same trajectory, so the motion is periodic in space.

Kinematics of Circular Motion

Circular motion is described by the angular quantities, which are the exact analogues of the translational ones met in Std XI.

$$\vec\omega = \dfrac{d\vec\theta}{dt}, \qquad \vec\alpha = \dfrac{d\vec\omega}{dt}$$

The tangential (linear) velocity is

$$\vec v = \vec\omega\times\vec r, \qquad v = \omega r$$

where $\vec r$ is the radius vector drawn from the centre of the circular motion.

Direction of the angular quantities

The direction of $\vec\omega$ is always along the axis of rotation and is given by the right-hand thumb rule: curl the fingers of the right hand along the sense of rotation with the thumb outstretched; the outstretched thumb gives the direction of $\vec\omega$.

Because the particle stays in the same plane, $\vec\alpha$ is also along the axis. For increasing speed $\vec\alpha$ is along $\vec\omega$; for decreasing speed it is opposite to $\vec\omega$.

If $\vec\alpha$ had a component perpendicular to the axis, it would change the direction of $\vec\omega$ as well, and since $\vec\omega$ is always perpendicular to the plane of rotation, the plane of rotation itself would change. In the special case where $\vec\alpha$ is constant in magnitude but always perpendicular to $\vec\omega$, it changes only the direction of $\vec\omega$ and never its magnitude — continuously changing the plane of rotation, exactly as an acceleration perpendicular to $\vec v$ changes only the direction of $\vec v$.

Period and frequency

$$\omega = \dfrac{2\pi}{T} = 2\pi n$$

with $T$ the periodic time in seconds and $n$ the frequency in rev/s.

Kinematical equations

When $\alpha$ is constant and along the axis, all the vectors are axial and scalar notation may be used.

$$\omega = \omega_0 + \alpha t, \qquad \theta = \omega_0 t + \dfrac{1}{2}\alpha t^2, \qquad \omega^2 = \omega_0^2 + 2\alpha\theta$$

One may equally work in revolutions rather than radians throughout (then $\theta$ is in rev and $\omega$ in rps), which removes $\pi$ from the arithmetic.

Uniform and non-uniform circular motion

Uniform circular motion (U.C.M.): during circular motion, if the speed of the particle remains constant, the motion is called uniform circular motion. Only the direction of the velocity changes, and the velocity is always tangential to the path. The acceleration responsible is the centripetal or radial acceleration

$$\vec a_r = -\omega^2\vec r, \qquad a_r = \omega^2 r = \dfrac{v^2}{r} = v\omega$$

which is constant in magnitude and always directed towards the centre — hence "centripetal". Illustration: any particle of a fan rotating uniformly. The value of the tangential acceleration in U.C.M. is always zero.

Non-uniform circular motion: when a fan is switched on or off, the speeds of its particles increase or decrease while their velocities stay tangential. The radial acceleration is still present, but its magnitude is no longer constant. The acceleration that changes the magnitude of the velocity is directed along or opposite to $\vec v$ and is therefore tangential — the tangential acceleration $\vec a_T$, with $a_T = \dfrac{dv}{dt}=r\alpha$. Since $v$ changes, $\omega$ changes too, which is the angular acceleration $\vec\alpha$.

For a body in non-uniform circular motion, the centripetal force is provided partly by the weight of the body and partly by the normal reaction.

Dynamics of Circular Motion

Centripetal force

The force providing the centripetal or radial acceleration is called the centripetal or radial force.

$$\vec F_{CP} = -m\omega^2\vec r$$

In magnitude,

$$F_{CP} = mr\omega^2 = \dfrac{mv^2}{r} = mv\omega$$

Its direction is along the radius and towards the centre (centre-seeking).

Four points the textbook insists on:

• The word centripetal is not the name or type of the force (as gravitational or nuclear are). It is an adjective describing its direction.
• While performing circular motion, the resultant of all the real forces acting on the body must be towards the centre; that resultant is what we call the centripetal force. Under its action the velocity stays tangential.
• The converse need not be true. A resultant force directed towards a centre does not always give circular motion — in S.H.M. the force is also always directed towards the centre of the motion. A matching tangential velocity is essential for the motion to be circular.
• The whole discussion is in an inertial frame, in which we observe the body performing circular motion.

Centrifugal force

Imagine sitting on a uniformly rotating merry-go-round with your eyes closed: you feel at rest. To explain being at rest in that rotating frame you must invoke a force equal in magnitude to the resultant real force but directed away from the centre,

$$\vec F_{CF} = +m\omega^2\vec r$$

This is the centrifugal force — a pseudo force arising from the centripetal acceleration of the frame of reference. It is non-real, but not imaginary: before the merry-go-round reaches uniform speed you really do experience an outward pull, and a force-measuring instrument records it. On reaching uniform speed, in the frame of the merry-go-round, the centrifugal force exactly balances the resultant of all the real forces, and the net force in that frame is zero. Only in such a frame may we say that centrifugal force balances centripetal force; there, "centrifugal force" means the net pseudo force and "centripetal force" means the resultant of all the real forces.

There are therefore two equivalent ways of writing the force equation for circular motion:

$$\text{Resultant of real forces} = m\omega^2 r \quad\text{(inertial frame)}$$

$$\text{Sum of real forces} + m\omega^2 r \;(\text{outward}) = 0 \quad\text{(rotating frame)}$$

Distinguish between centripetal force and centrifugal force

Standard Results

• $\vec v=\vec\omega\times\vec r$, $\quad v=\omega r$, $\quad \omega=\dfrac{2\pi}{T}=2\pi n$
• $a_r=\omega^2r=\dfrac{v^2}{r}=v\omega$ (towards the centre); $\quad a_T=r\alpha$ (tangential); $\quad a=\sqrt{a_r^2+a_T^2}$
• In U.C.M., $a_T=0$
• $F_{CP}=mr\omega^2=\dfrac{mv^2}{r}=mv\omega$; $\quad \vec F_{CP}=-m\omega^2\vec r$; $\quad \vec F_{CF}=+m\omega^2\vec r$
• $\omega=\omega_0+\alpha t$, $\quad \theta=\omega_0t+\dfrac{1}{2}\alpha t^2$, $\quad \omega^2=\omega_0^2+2\alpha\theta$ (constant axial $\alpha$ only)