Question

How does torque differ from force?

Answer 1

Force is a vector (applied in a straight line, with the acceleration in the same direction as the net force).

Torque `\vec tau` refers to applying a force `\vec F` to a lever arm at a specific distance `\vec r`from the center of rotation.

`\vec tau = \vec r \times \vec F`

You push your car with a force, but you apply torque to the lug nut to loosen it in order to change your tire. With torque, you talk about force at a distance, and angular acceleration instead of linear acceleration.

Force has units of Newtons, while torque, being the product of a force and a distance, has units of Newton`\cdot`meters.

Torque and force also differ in that force is a true vector, whereas torque is a pseudovector, that is, it picks up a sign flip under certain kinds of coordinate transformations.

In terms of derivatives, force is the time derivative of the linear momentum:
`\vec F = \frac{d \vec p}{dt}`
while torque is the time derivative of angular momentum:
`\vec \tau = \frac{d \vec L}{dt}`