If the body rotates through equal angles in equal time intervals, we say that it rotates with uniform angular velocity.
If this is not the case, the body is rotationally accelerated.
The angular acceleration is defined as
Now, if we switch off the fan, the angle rotated in any second is smaller than the angle rotated in the previous second. The angular velocity
decreases as time passes, and finally it becomes zero when the fan stops. The fan has an angular deceleration.
But then why do we need to switch on the fan in order to start? If an angular acceleration may be achieved with zero external force, why doesn't a wheelchair start moving on the floor as soon as one wishes it to do so? Why are we compelled to use our muscles to set it into motion?
In fact, one cannot have angular acceleration without external forces.
Then, what is the relation between the force and the angular acceleration?We find that even if the resultant external force is zero, we may have angular acceleration. We also find that without applying an external force, we cannot have an angular acceleration. What is responsible for producing angular acceleration?
The answer is Torque.Moment of a force
Let us consider a single particle, whose position with reference to the origin O of the two-dimensional coordinate system is given by position vector
. The location of the particle is P (x, y). Let q be the angle between OP and X-axis.
Suppose a force
applied on the particle changes its position from P to Q. If PQ is a short distance on the circumference of a circle of radius r, then OP = OQ = r.

As the particle moves from P to Q, the distances x and y change by Dx and Dy respectively. Now PQ = rDq
If Dx is the projection of 'rDq' in the X-direction,

Similarly, Dy is the projection of 'rDq' in the Y-direction.

Suppose a force
is applied at P (x, y), then the work done by the applied force is given by



The quantity t is known as the torque of the applied force about an axis which passes through O and is perpendicular to the X-Y plane. The torque measures the turning or twisting effect of force. The torque is called as the moment of force.
Expression for force can be written in terms of angles also.








[
sin (A - B) = sin A cos B - cos A sin B]
It is clear from the figure that,

Here, d is the perpendicular distance of the line of action of the force from the origin O. This is known as the lever arm of the force about the origin O.


