# Torque, Angular Acceleration and Linear Acceleration

We know that we torque is applied, it cause an angular acceleration in the rotating body similar to what a force does to a body moving on a straight line.

But my question is, Does Torque affect the linear acceleration of the rotating body along with the angular acceleration?

We know from the formula that

Angular Acceleration = Perpendicular dist. × Linear Acceleration

Torque = Perpendicular Distance × Force

So can it be said that the body gets a linear acceleration as well?

(Consider gravity free situation, single force (i.e only that force which causes torque) and no friction.)

• It depends on the forces acting on the body. Are you just considering a single force? You can't say in general if there is a net torque if there will be a net force as well. – Aaron Stevens Jan 29 at 15:54
• @AaronStevens I have made changes for what to consider. Please check. – Asad Ahmad Jan 29 at 15:59
• @AsadAhmad Aaron Stevens pointed out a flaw in my original answer. Please see my revised answer. – Bob D Jan 29 at 17:38

## 2 Answers

But my question is, Does Torque affect the linear acceleration of the rotating body along with the angular acceleration?

Although the applied force always stays perpendicular to the vector , at each instant there is a net force acting on the body and therefore there will always be some translational motion and acceleration. See left diagram below.

The only sure way to have only rotation without translation is to a apply pure force couple (two equal and opposite parallel forces). It will induce pure rotation without translation. See right diagram. Hope this helps.

Hope this helps. • @AaronStevens Good point. I guess I was thinking of the force instantly changing so that it was always perpendicular to the displacement vector. As a practical matter this is probably not achievable. The only sure way to prevent translation is the force couple of the right diagram. Space ships probably have thrusters of the type in the right diagram to insure rotation without translation. I will revise my answer. Thanks. – Bob D Jan 29 at 17:26

You are essentially wondering if we can say anything about the net force if we know what the net torque about some point is. In general this question cannot be answered. You can have scenarios where we have a net torque with no net force, scenarios where we have a net force with no net torque, and anything in between.

Consider gravity free situation, single force (i.e only that force which causes torque) and no friction.

In this case there must be a net force. This is because with only one force the net force is just this force.$$^*$$ This is independent of the torque of this force about some point. Moving the application point of the force could change the torque the force has about some point, but the net force will remain the same.

$$^*$$As a sort of "proof by contradiction" of this point, in order for the net force to be $$0$$ then must mean the applied force is also $$0$$, which is a contradiction of saying we apply a force. Therefore, in only applying one force we know there is a non-zero net force.

• Why are you considering the net force to be zero? If a single force is tangentially applied, there is a torque. That is the scenario. Even considering the centripetal force, the net force is not zero in this case. So, shouldn't there be any tangential acceleration due to the force that causes the torque? – Asad Ahmad Jan 29 at 16:30
• @AsadAhmad I am not saying the net force is $0$. I said the net force cannot be $0$ for a single force, and used a sort of "proof by contradiction" to show why this is the case – Aaron Stevens Jan 29 at 16:51