Question

Two particles having mass M and m are moving in a circular path having radius R and r. If their time period are same then the ratio of angular velocity will be

• A. $ \frac{r}{R}$
• B. $ \frac{R}{r}$
• C. 1
• D. $\sqrt{ \frac{R}{r}}$

Answer 1

The Correct Option is C

Let radius for the particle of mass M = OB = r and for the particle of mass m = OC = R 
Let linear velocity for a particle of mass M = v₁ and for the particle of mass m = v₂ 
Let angular velocity for the particle having mass M = and for the particle having mass m= 
Let Time period for the particle having mass M = T₁ and for the particle having mass m = T₂  
\(T_1 = \frac{2\pi r}{v_1} \quad \text{and} \quad T_2 = \frac{2\pi R}{v_2}\)
Given: T₁ = T₂
\(⇒\)\(\frac{2\pi r}{v_1} = \frac{2\pi R}{v_2}\)
\(⇒\)\(\frac{r}{v_1} = \frac{R}{v_2}\)
\(⇒\)\(\frac{v_1}{r} = \frac{v_2}{R}\)

The above equation generated is the formula for angular velocity. hence: 
\(⇒ \omega_1=\omega_2\)
\(⇒\frac{\omega_1}{\omega_2}=\frac{1}{1}\)
Therefore, the ratio of the angular velocity will be 1:1.
Therefore, the correct option is (C) : 1.