Definition of Angular Velocity

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Angular velocity of a rotating rigid body is the rate of change of angle swept.

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

When we switch on the fan, the fan rotates about a vertical line, the angle rotated by the fan in the 1^st second is small, that in the 2^nd second is larger, that in the 3^rd second is still larger and so on. The fan, thus has an angular acceleration. The angular velocity increases with time. If we wait for a couple of minutes, the fan attains constant speed. The angle rotated in any time interval is now equal to the angle rotated in the successive equal time interval. Now the fan is rotating uniformly about the vertical axis.

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.

When one switches a fan on, the centre of the fan remains unmoved while the fan rotates with an angular acceleration. As the centre of mass remains at rest, the external forces acting on it must add up to zero. This means that one can have angular acceleration even if the resultant external force is zero.

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 forceapplied 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.