Question

A body is whirled in a horizontal circle of radius 20 cm. It has an angular velocity of 10 rad/s. What is its linear velocity at any point on a circular path?
a. \[20\,m/s\]
b. \[\sqrt 2\,m/s\]
c. \[10\,m/s\]
d. \[2\,m/s\]

Answer 1

Hint:When a body is moving in a circular path then the position of the body along the circular path changes with time. As we know, the rate of change in the linear position of the body with respect to time is called linear velocity. The rate of change of the angular position with respect to time is called the angular velocity.

Formula used:
\[v = \omega r\]
where v is the linear velocity, \[\omega \] is the angular velocity and r is the distance of the point from the axis of rotation.

Complete step by step solution:
It is given that the body is whirled in a horizontal circle. The radius of the circle is given as 20 cm. The angular velocity of the body is given as 10 rad/s. We need to find the linear velocity at any point on a circular path.

Image: Body motion in a circular path

Let the linear velocity of the body is v. The distance of the body from the axis of rotation, i.e. from the center of the circular path is equal to the radius of the circular path.
\[r = 20\,cm = 0.2\,m\]
\[\Rightarrow \omega = 10\,rad/s\]
When a body is moving in a given circular path then the position of the body along the circular path changes with time.The rate of change in the linear position of the body with respect to time is called the linear velocity, v.

The linear velocity of the body is linearly related to the angular velocity of the body in the circular path as,
\[v = \omega r\]
Putting the values, we get the linear velocity of the body in circular orbit as,
\[v = 10 \times 0.2\,m/s\]
\[\therefore v = 2\,m/s\]
Hence, the linear velocity of the body is equal to 2 m/s.

Therefore, the correct option is d.

Note: We should be careful while using the relation between the linear velocity and the angular velocity as it is for the instantaneous velocity. So, if we find the linear velocity of the accelerated body in a circular path then the obtained linear velocity will be corresponding to the angular velocity at that instant in time.