0.32 Phy1320: angular momentum -- the mathematics of torque  (Page 9/13)

How are force and torque related?

To begin with, torque is proportional to the magnitude of the applied force, but that is not the end of the story. The rest of the story involvesexactly where and in what direction the force is applied.

Create a torque to close a door

For example, consider applying a force for the purpose of creating a torque to close an open door. Initially, the door has zero angular velocity.

You could apply a force in any direction at any point on either side or the edge of the door that youare tall enough to reach. However, with regard to the objective of closing the door, it would matter very much where and in what direction you applied theforce.

A very intuitive topic

The interesting thing about this topic is that you already know all about it from a practical and intuitive viewpoint. You would already know intuitivelywhere and in what direction to push on the door to cause it to close with a minimum expenditure of energy.

You probably wouldn't push on the edge of the door

If you pushed on the edge of door, directing your force directly at the hinges (the axis of rotation), the door wouldn't move. While this might prove to be a good form ofisometric exercise, it would not be an effective way to close the door.

A force in any other direction

A force that is applied to the door, (even on the edge of the door) acting in any direction other than directly toward the hinges could be decomposed into two components:

• A radial component acting directly towards the axis or rotation or the hinges.
• A perpendicular component acting perpendicular to the surface of the door.

The radial component is wasted effort

The radial component would make no contribution to the development of the torque required to change the angular velocity of the door and cause the door to close.Only the perpendicular component would contribute to the development of such a torque.

Could develop torque at any point on the door

So now you know that you could apply a force at any point on the door, and so long as that force has a component that is perpendicular to the surface of thedoor, the perpendicular component would contribute to the development of torque.

Location, location, and location

However, it is also important where you push on the door to apply the force. If you push on what we normally consider to be the inside surface of thedoor, it might create a torque, but that torque may have the wrong sign or direction to cause the door to close. In fact, that would cause the door to openeven further.

Once you realize that you must push on what we would call the outside surface of the door, it would still be important where you push and apply the force.Suppose for example that you were to push at a point that is only one inch away from the hinge. You know intuitively that even for a lightweight door, you mighthave to apply a very strong force to cause the door to close by applying the force at that location.

Where would you push?

You would probably push on the door at a point somewhere between the center of the doorand the outer edge of the door.