Torque

Torque is defined as the product of radius, force, and the sine of the angle between force and radius:

t = r F (sinq)
figure 1

Here, radius is measured from the pivot point (center of rotation), and points away from the pivot when it is considered as a vector.

Torque is the agent of rotation: when a nonzero net torque is applied to an object, an angular acceleration around the pivot is produced. Torque can be large when force is large, but leverage is also a consideration: even a small force can produce a large torque on something if the force is applied at right angles to the radius and as far from the pivot point as possible (ie. t is large when q is 90o and r is as big as possible).

The unit of torque is the Newton-meter (Nm). Torque is considered to be a vector when the cross-product definition is used:

t = r x F

which has magnitude rFsinq, and direction given by the right-hand-rule (curl fingers of right hand from direction of r toward direction of F; thumb points direction of t). The torque vector points toward you in figure 2, but away from you in figure 1. Try experimenting with the vector nature of the cross product with this Java Applet. One can also say that when the torque vector points toward you, it has a "counterclockwise" tendency and when it points away, it has a "clockwise" tendency.

Example: calculate the torque on this stick which has a hinge at its top end, and a force pulling on it as shown.

The distance along the stick from the pivot point (hinge) to the point of application for the force is r = (2.5m) - (0.8m) = 1.7 m. The force makes a known angle with the horizontal, and the stick is vertical, so we have to use geometry to calculate q = 90o - 20o = 70o. Then, applying the torque equation we get t = (1.7m)(20N)(sin70o) = 31.9 Nm.
figure 2

If you need help figuring out the various angles and relating them to moment arm, try the trigonometry tutorials #5 and #6 at Math Help.

Moment Arm

Moment arm is defined as the perpendicular distance from the pivot to the line-of-action of the force. It is also called lever arm in our lab manual. Moment arm is a pretend value of radius: it represents the "effective radius" that a force has to work with when the force is not perpendicular to radius. Moment arm is always less than or equal to radius.

In the previous example, the force is applied at a 70o angle to the stick. The radius is effectively less than 1.7 m; the system behaves like it has an effective radius of (1.7m)(sin70o) or 1.6m (slightly smaller). So moment arm r^ = r sinq, and the torque equation can be rewritten as:

t = r F sinq = r^ F.

The physical position of moment arm is obtained by the following procedure:

  1. Extend the line-of-action of the force in both directions.
  2. Lay a straightedge on this line, then rotate the straightedge 90o.
  3. Translate the perpendicular straightedge along the line until it intersects with the pivot (or hinge).
  4. Draw a perpendicular line segment from the pivot to the line-of-action, and label this r^.
There should be a triangle formed with r as the hypotenuse, r^ as one of the shorter sides, and an angle in the triangle that can be determined by plane geometry.

Using the example above, the moment arm is formed by extending the line of the pulling force to the left, then drawing the red perpendicular line segment shown in this figure.

You can show that the angle inside the triangle is 70o, and that r^ = r sin70o.
Here is another example: the bicep muscle pulls up on the human forearm to flex the joint, making an angle qm with the horizontal. The moment arm is a fraction of the already-small distance from the pivot (elbow) to where the muscle attaches to the bone.

The muscle must exert a huge force because of its small leverage, in order to produce a torque on the arm.

On to the exercises...