Question

Two cars having mass ${m_1}$ and ${m_2}$ move in circles of radio ${r_1}$ and ${r_2}$ respectively. If the two cars complete their circle in equal time. Then, find the ratio of their angular speeds $\dfrac{{{w_1}}}{{{w_2}}}$.
(A) $\dfrac{{{m_1}}}{{{m_2}}}$
(B) $\dfrac{{{m_1}{r_1}}}{{{m_2}{r_2}}}$
(C) $1$
(D) $0$

Answer 1

Hint: As the cars are moving in the circle and complete their circle in equal time. So, male the time of both the cars of completing the circle same and by using the formula of angular velocity find their ratio.
Formula used The formula which will be used for finding the angular velocity is –
$\omega = \dfrac{\theta }{t}$
where, $\omega $ is the angular velocity
$\theta $ is the total angle, and
$t$ is the time of completing the journey

Complete step by step solution:
The angular speed of an object is calculated when the object is in the circular motion. So, the circular motion can be defined as the type of motion in which the object follows the circular path. The motion of the body moving with constant speed along the circular path is called uniform circular motion. So, here, the speed is constant and velocity changes.
Now, the angular velocity can be defined as the velocity of the object produced by the object which is in circular motion. It is denoted by the $\omega $. As the total distance covered in the circle is equal to the total angle covered by it so, the angular velocity can be represented as –
$\omega = \dfrac{\theta }{t}$
We know that, total angle of circle is equal to $2\pi $
$\therefore \omega = \dfrac{{2\pi }}{t} \cdots \left( 1 \right)$
Let the time of completing the circle of both the cars be $t$ because they both complete the circle at the same time. Now, according to the question, it is given that there are two cars of mass ${m_1}$ and ${m_2}$ move in circles of radio ${r_1}$ and ${r_2}$.
Using the equation $\left( 1 \right)$
Angular velocity of first car, ${\omega _1} = \dfrac{{2\pi }}{t}$
Angular velocity of second car, ${\omega _2} = \dfrac{{2\pi }}{t}$
Now, we have to calculate the ratio of both the angular velocities –
$
  \therefore \dfrac{{{\omega _1}}}{{{\omega _2}}} = \dfrac{{\dfrac{{2\pi }}{t}}}{{\dfrac{{2\pi }}{t}}} \\
   \Rightarrow \dfrac{{{\omega _1}}}{{{\omega _2}}} = 1 \\
 $
Hence, the required ratio is 1.

So, the correct option is (C).

Note: The angle covered by the circle is equal to ${360^ \circ }$ which can also be written as $2\pi $. Now, we know that velocity is the ratio of speed to the time so, as the angular velocity is the velocity of a particle in circular motion then, the distance covered by it will be equal to $2\pi $.