Question

Two objects of masses '$m_1$' and '$m_2$' are moving in the circles of radii '$r_1$' and '$r_2$' respectively. Their respective angular speeds '$\omega_1$' and '$\omega_2$' are such that they both complete one revolution in the same time '$t$'. The ratio of linear speed of '$m_2$' to that of '$m_1$' is

• A. $\omega_1 : \omega_2$
• B. $T_2 : T_1$
• C. $m_1 : m_2$
• D. $r_2 : r_1$

Answer 1

Answer: (D)

$r_2 : r_1$

Answer 2

The linear speed $v$ in uniform circular motion is given by

$$v = \omega r$$

Since both objects complete one revolution in the same time $t$, their angular speeds are:

$$\omega_1 = \omega_2 = \frac{2\pi}{t}$$

Thus,

$$v_1 = \omega_1 r_1 \quad \text{and} \quad v_2 = \omega_2 r_2$$

The ratio of the linear speeds is:

$$\frac{v_2}{v_1} = \frac{\omega_2 r_2}{\omega_1 r_1} = \frac{r_2}{r_1}$$