# Does Torque produce acceleration of the particles that move in a circle about the axis of rotation?

If torque is applied to an object, it begins to rotate.

1) During rotation, every particle moves in a circle about the axis of rotation. Do these particles accelerate while executing this circular motion after torque is applied to it? In the derivation of the rotational kinetic energy and the moment of inertia of a rotating rigid body, we consider particles not accelerating but each moving with different velocities. But for Torque, a force has to be applied... and so shouldn't this force cause an acceleration(linear) of the particles?

2) My second question is completely unrelated to this-

Consider a thin (uniform) rod rotating about an axis passing through its end and perpendicular to its length. Does the rod rotate about its centre of mass or about the axis of rotation?

A similar question had been asked and this is part of the answer: " A pure torque any point on the body (with no net force) will purely rotate a rigid body about its center of mass."

Since a torque has to be applied to cause rotational motion, why not say that the rod rotates about the axis of rotation? Why the COM of the rod?

3)My third question is that if the rotation is about the COM, does the COM remain stationary/ move with a constant velocity?

Torque is actually an term given to effect of force if the Force does not pass through the axis of rotation. So a force can produce a rotation along with linear acceleration. For example motion of a rolling tyre, the friction produces rotation as well as linear acceleration. As far as a couple is considered it only produces a rotation and not an linear acceleration.

• I think what I meant to ask was this: During rotation, every particle moves in a circle about the axis of rotation. Do these particles accelerate while executing this circular motion? – Gokulakrishnan Shankar Mar 3 '18 at 16:42
• That's exactly what I answered: Like in the case of rolling of a car tyre any point on the tyre (disc) has both an angular as well as an linear component of acceleration. – Burhaan u din Mar 4 '18 at 12:06

The COM only moves if there is a net torque on the body, and the body on changes its moment of inertia if there is a net torque.

Linear acceleration should never be proportional to torque. Let's imagine it is:

$${\bf \vec a} = k{\bf\vec{\tau}}$$

$$\frac{d^2 {\bf \vec x}}{dt^2} =k({\bf \vec r \times \vec F})$$

Now take the mirror image (so vectors change sign):

$$\frac{d^2 (-{\bf \vec x})}{dt^2} =k(-{\bf \vec r} \times -{\bf \vec F})$$

$$-{\bf \vec a} = k{\bf \vec{\tau}}$$

So unless $k=0$, Parity symmetry is violated.

• I'm sorry... What I meant to ask was this: During rotation, every particle moves in a circle about the axis of rotation. Do these particles accelerate while executing this circular motion? I edited the question just now.... – Gokulakrishnan Shankar Mar 3 '18 at 17:44