Torque

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We learned that force was the key factor that caused acceleration hence motion in objects. What causes an object to roll? This question is answered by a special force called, torque.

Eq 1: t = r x F

Because force F and distance r are vector quantities, r x F is the cross product.
Using the definition for the cross product and transforming the vectors to its magnitude:
Eq 2: t = rFsin q where q is the angle between r and F.

And now to relate torque to the mechanics of spinning, I is the moment of inertia and a is the angular acceleration.
Eq 3: t = Ia

Simply analyzing equation 2, it shows that at angle q = 90, the maximum torque is obtained. And also, torque is directly related to the radius and the force. This direct relationship is analogous to a famous saying by Aristotle, “If you give me a long enough board I will lift the earth.” Aristotle was using the fact that the lever system pushes stronger as the distance between the force and the fulcrum increases. It is like seesaw, where two weights can be balanced by varying the distance at which the two weights are placed. Another example can be noticed in tools. A wrench produces torque by increasing the length of its handle. A small force applied to the end of the handle will give great torque to turn.

Now, after completely understanding what torque means, we need to apply it to the mechanics of pool. To possess a complete control of the cue, one must know how the cue spins, how much it spins and what must be done to create the revolution on the cue ball. Again, using the equation 2, we realize that applying the force at the very edge of the ball, where the angle created by the force and the distance from the center of the ball is perpendicular maximizes the torque.

 

The opposite effect of this, is when force is applied in the center of the cue. However great the force, the distance from the center will always be zero thus no initial angular acceleration will result. The phrase, initial angular acceleration was used because after the ball begins to translate (no rotation), it begins to spin. And the cause of this is analyzed in the next section, Friction.

Sample Problem
I hit the cue ball on its very edge with force = F at angle = q (theta) from the horizontal. The resulting linear acceleration of the cue was a. What was the resulting angular acceleration of the cue? (Radius = R)

Your Answer:

Solution