Two particles of mass M and m are moving in a circle of radii R and r. If their time-periods are same, what will be the ratio of their linear velocities

The correct option is R : r.

To find the ratio of the linear velocities of the two particles, we can analyze the relationship between linear velocity, radius, and the time period of circular motion.

The linear velocity ($v$) of a particle moving in a circle of radius ($r$) with a time period ($T$) is given by the formula:
$v=\frac{2\pi r}{T}$
where $2\pi r$ represents the circumference of the circular path (the distance covered in one complete revolution).

Let $v_1$ be the linear velocity of the first particle of mass $M$ moving in a circle of radius $R$ with time period $T$:
$v_1=\frac{2\pi R}{T}$
Similarly, let $v_2$ be the linear velocity of the second particle of mass $m$ moving in a circle of radius $r$ with the same time period $T$:
$v_2=\frac{2\pi r}{T}$

Now, we find the ratio of their linear velocities ($v_1:v_2$):
$\frac{v_1}{v_2}=\frac{\left(\frac{2\pi R}{T}\right)}{\left(\frac{2\pi r}{T}\right)}$
Simplifying the expression by cancelling the common terms ($2\pi$ and $T$) in both the numerator and the denominator, we get:
$\frac{v_1}{v_2}=\frac{R}{r}$

Therefore, the ratio of their linear velocities is $R:r$.