If ratio of centripetal acceleration of two particles moving on the same path is 3 : 4. Find the ratio of their tangential velocities.

To find the ratio of the tangential velocities of two particles moving on the same path, we start by analyzing the formula for centripetal acceleration.

The centripetal acceleration $a$ of a particle moving in a circular path of radius $r$ with a tangential velocity $v$ is given by the formula:
$a=\frac{v^2}{r}$

Since both particles are moving on the same path, the radius of curvature $r$ of the path is the same for both particles (i.e., $r_1=r_2=r$).

Let $a_1$ and $a_2$ be the centripetal accelerations of the two particles, and let $v_1$ and $v_2$ be their respective tangential velocities. The ratio of their centripetal accelerations is:
$\frac{a_1}{a_2}=\frac{v_1^2/r}{v_2^2/r}=\frac{v_1^2}{v_2^2}$

We are given that the ratio of their centripetal accelerations is 3 : 4:
$\frac{a_1}{a_2}=\frac{3}{4}$

Substitute this ratio into our equation:
$\frac{v_1^2}{v_2^2}=\frac{3}{4}$

To find the ratio of their tangential velocities, we take the square root of both sides of the equation:
$\frac{v_1}{v_2}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$

Therefore, the ratio of their tangential velocities is √3 : 2.