If this dependence on angle isn't apparent to you, consider your answer
to the fourth introductory exercise. Would you try to loosen a bolt
by pushing along the direction of the wrench? You wouldn't be very
successful if you did! The bolt wouldn't budge a bit, because you
would be pushing the wrench against the bolt rather than twisting. The
torque depends only on the component
to . (Remember that
is the line pointing from the pivot point to the point at which the force
is applied.) The components of
are illustrated below.
As the angle between the applied force and the distance to the pivot point decreases from 90o to 0o, the torque decreases from its maximum value to zero. Mathematically, we see
Does this equation look similar to anything you've seen before? and are both vector quantities. Do you remember the ways we can multiply two vector quantities? One of those methods, the cross product, involves the sine of the angle between the two vectors being multiplied:
(To review the properties of the cross product, go to one of the vector tutorial sites.) Notice the magnitude signs The cross product of two vectors results in a third vector. The magnitude of this product vector is given by the right-hand side of the above equation. But this has the exact same form as the expression for torque! Indeed, torque is the cross product of distance and force:
Since torque is the cross product of two vectors, it is a vector itself. The direction of the torque can be found by the right-hand rule for cross products: put your fingers in the direction of the first vector () and your palm facing in the direction of the second vector (); your thumb will point in the direction of the cross product (). For the see-saw of Sample Probem 1, pointed from left to right, and the force was toward the bottom of the screen. As shown in the figure, use of the right-hand rule results in the thumb pointing into the page.
Thus would point into the page in Sample Problem 1.
Often, however, we want torque for more than one force. And sometimes
we don't know all of the forces acting on an object. The right-hand rule
can also relate the direction of the torque to the direction of rotation.
Consider again the see-saw of Sample Problem 1. The gravitational force
acting to the right of the pivot point causes the see-saw to turn in a
clockwise direction. Look at the hand in the figure above. If the fingers
were to curl, they would curl in the direction of rotation. Thus, if you
curl the fingers of your right hand in the direction of rotation, your
thumb will point in the direction of torque.