# Why does torque lead to rotation? [duplicate]

I know that torque will cause an object to rotate about its centre of mass. I am curious why this is so.

Suppose we have a string of particles that are held in a line by electromagnetic forces of attraction such that they resemble a rod. The rod is floating in space. If we were to apply a perpendicular force at one end of the rod, it should start rotating about its centre of mass.

As such: Why does the object rotate about its centre of mass? I know that this is intuitive if there is a pivot supporting the centre of the rod. However, in this case there isn't, yet the object still behaves as though it is pivoted at the centre.

Why are the particles at both ends of the rod accelerating in opposite directions? If a perpendicular force was applied at one end of the rod, shouldn't the rod experience translational motion, as though it is being dragged forward by the end where the force was applied?

More specifically, I would like to know how the individual particles in the rod are "pulling and pushing" each other when the rod is rotating.

• – sammy gerbil Sep 24 '18 at 13:06
• – sammy gerbil Sep 24 '18 at 14:11

This is a very good question. One important consideration is that when you push one end of the rod, all of the particles in the rod do not move all simultaneously. There is a very slight delay (which is represented physically by a slight bending in the rod when you torque it, even when there isn't a pivot).

The particle you are applying the force to will move in the direction of the applied force. This motion

1. causes this particle and the particle next to it to separate slightly and
2. causes the line connecting together the two particles to slant slightly compared to the rest of the rod.

The interparticular forces keeping the rod as rigid as possible acting between these first two particles will now increase in strength due to (1) and have a component of force perpendicular to the rest of the rod due to (2). Since there is now a component of force acting on the second particle perpendicular to the rod, it starts to accelerate in the same fashion as the first. This trend continues down the whole rod.

In practice, this process is extremely fast for rigid rods, but is visible for something softer like a pool noodle.

I know that torque will cause an object to rotate about its centre of mass. I am curious why this is so.

If the net external forces on an object are zero, then the center of mass of an object will experience no acceleration: $$F_{net} = ma = 0$$. With no acceleration, we can always find an inertial frame where the center of mass of the object is at rest. Restated, any rotation that the object experiences does not cause the center of mass to move. Therefore if there is a rotational center, it must coincide with the center of mass.

Why are the particles at both ends of the rod accelerating in opposite directions?

At the moment the force is applied, this does not happen. The rod doesn't start rotating with the center of mass stationary, with one end moving left and the other end moving right. It starts rotating and moving at the same time. The combination of these two motions means the end where the force is applied accelerates the most quickly, while the other end accelerates (in the same direction) more slowly. Looking at it from the side, you might see one end moving quickly right, while the other end is nearly stationary, but slowly starts moving to the right as well.

If a perpendicular force was applied at one end of the rod, shouldn't the rod experience translational motion, as though it is being dragged forward by the end where the force was applied?

Yes. Any net force on an object causes acceleration of the center of mass. So off-center forces cause linear acceleration and torque (rotation) simultaneously.