Question

The angular velocity of a particle rotating in a circular orbit 100 times per minute is:
A. 1.66 rad/s
B. 10.47 rad/s
C. 10.47 deg/s
D. 60 deg/s

Answer 1

Hint: The angular velocity of the body in a circular motion is the rate of change of angular position with respect to time. The S.I. unit of the angular velocity is radians per second.

Formula used:
\[\omega = \dfrac{{\Delta \theta }}{{\Delta t}}\],
Here \[\omega \]is the angular velocity, \[\Delta \theta \]is the angular displacement in time \[\Delta t\].

Complete step by step solution:
It is given that the particle is rotating in a circular orbit. The particle completed 100 revolutions around the circular orbit each minute. When a particle completes one complete revolution along the circular path then the angular displacement is equal to \[2\pi \] radian, i.e. 360 degrees. So, the total angular displacement in 100 complete revolutions will be,
\[\Delta \theta = 100 \times 2\pi \] radians
\[\Rightarrow \Delta \theta = 200\pi \] radians.

The time of the journey is 1 minute to complete 100 revolutions. As we know that angular velocity is the angular displacement per unit of time. As per the S.I. unit, the unit of time in seconds. So, we need to change the given time of journey in the S.I. unit.
\[\Delta t = 1\min \]
\[\Rightarrow \Delta t = 60\sec \]
Using the angular velocity formula, the angular velocity of the given particle will be,
\[\omega = \dfrac{{\Delta \theta }}{{\Delta t}}\]
\[\Rightarrow \omega = \dfrac{{200\pi }}{{60}}\,rad/s\]
\[\therefore \omega \approx 10.47\,rad/s\]
Hence, the angular velocity of the particle is approximately 10.47 rad/s.

Therefore, the correct option is B.

Note: We should change the given data regarding angular displacement in terms of radian and time in seconds to find the angular velocity in the standard form. If the question is specifically asked to calculate the angular velocity in terms of degree/sec then we change the angular displacement in degrees and time in terms of the second. The other frequently used unit of angular velocity is rpm, which stands for revolution per minute.