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Torque

Written by tutor Alexandra L.

Torque is often described as the analog of force in angular motion. More precisely, it is a quantity that describes the ability of a force to cause twisting or turning of a body - that is, to produce angular motion. Intuitively, we may think of applying a force of constant magnitude (such as a push) to a door at several different distances away from its hinges - the farther away from the hinges we push, the more easily the door swings shut. If we apply the force very close to the hinges, we must push harder in order to achieve the same turning effect.

Algebraically, then, the turning effect is directly proportional to both the magnitude of the force, and the perpendicular distance between the force vector and the hinge (more generally termed the fulcrum) around which angular motion is to occur. This perpendicular distance is often called the lever arm, or the moment arm of the force vector about the fulcrum.

Torque representation

In terms of vector algebra, the relation between torque, force, and lever arm is the cross product of the force vector and the lever arm (whose direction points from the fulcrum to the point of force application):

Torque equation

In terms of the angle φ between the force vector and the lever arm vector, the magnitude of the torque is given by:

Magnitude of Torque equation

The direction of the resultant torque is given by the right hand rule - when the fingers of the right hand curl from the direction of r to the direction of F (that is, in the direction of rotation produced by the torque), the thumb will point in the direction of the torque vector τ. By convention, counterclockwise torques are positive, while clockwise torques are defined as negative.

Torque and angular acceleration

The angular acceleration of a rigid rotating body depends on the total amount of torque about the axis of rotation. This provides us with a formula analogous to Newton’s second law for forces that describes the rotational motion of rigid objects:

Στ =

The analog of mass in linear motion is I, the moment of inertia of the object.

Torque Practice Quiz

True or False? If you grasp the handle of a spanner closer to the nut you are trying to tighten, you will need to apply less force to tighten it.

A. True
B. False
The correct answer here would be B.

True or False? Maximum torque is produced when the force vector is perpendicular to the lever arm connecting the fulcrum and the point of force application.

A. True
B. False
The correct answer here would be A.

Two children are sitting on a see-saw. The first child is twice as heavy as the second child, and is sitting a distance of 1.0 m away from the middle of the see-saw. How far from the middle of the see-saw (on the other side) must the second child sit in order to balance out the see-saw?

A. 1.0 m
B. 0.5 m
C. 2.0 m
The correct answer here would be C.
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