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Moment of a Force: Torque

The simplest example of rotatory motion is the opening or closing of a door. When you apply a force at the handle A, we will notice that you can close the door with very little effort.

If you apply a force at a point B, somewhere midway between the handle and the hinge C, you will have to put in a much greater effort to close the door. Next try to close the door by applying a force at the hinge C.

You will not be able to close the door even if you apply an extremely large force. Thus we conclude that in rotatory motion, the effect of force depends on the magnitude of the applied force and the point of application of the force.

When the force is applied at point A, the distance between the point of application of force and axis of rotation which is the line PCQ is AC
In the second it is BC, and in the third (when the force is applied at C), zero.

Since AC > BC >0, we conclude that the rotatory effect is more if the distance between the point of application of force and the axis of rotation is larger.

Further, in each case, the rotatory effect will be more if the magnitude of force is larger. The third factor is the direction in which the force is applied. The direction in which the force acts is called the line of action of the force.

If you apply a force at the handle A of the door in a direction perpendicular to the face of the door, it is much easier to close the door than when the force is applied at an angle. In particular, if the line of action of the force applied at A is parallel to the face of the door, no rotation is possible.

Thus we conclude that the rotatory effect of a force on a body, which is capable of rotation an axis, depends upon
  1. the magnitude and the direction of the force, i. e. the factor vector, and
  2. the axis of rotation. 
The rotatory effect of a force is characterized by what is called torque (or moment of force) which is the analogue of force in translatory motion.

Torque acting on a Single Particle

Consider a particle P whose position vector with respect to origin O of an inertial reference frame is r. 

If a force F acts on the particle, as shown in the figure, the magnitude of torque acting on the particle with respect to the origin O is defined as
|| = |r| |F| sin θ
or simply τ = r F sin θ
where θ is the angle between vectors r and F.
In vector notation, 
= r × F
The product r × F is called the vector (or cross) product of vectors r and F which is defined as the product of the magnitudes of the two vectors and the sine of the angle between them. The cross product of two vectors is a vector.
 
Torque is a vector quantity. Its magnitude is given by = r F sin θ.
Its direction is normal to the plane containing vectors r and F and can be determined by the right-handed screw rule.
 
According to this rule, if we curl the fingers of our right hand in the direction in which vector r must be rotated to move into the position of vector F through the smaller angle between them, the extended (erect) thumb gives direction or sense of torque .

Unit of Torque

Torque has the same dimensions as those of work (both being force times distance) viz. ML2T-2
The two are, however, very different quantities. 
Work is a scalar, torque is a vector.

To distinguish between the two we express work in joules and torque in Newton-metre (N m).

Dependence of Torque on the Moment Arm
We observe that the torque produced by a force depends not only on the magnitude and the direction of the force (i. e. the force vector F) but also on the point of application of the force relative to the origin O (i. e. on vector r). 

In particular, when particle P is at the origin O, so that the line of application of force passes through the origin, r is zero and the torque about the origin is zero.
The magnitude of as
τ = (r sin θ ) F = rF
or as τ = r (F sin θ ) = r F




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