Question

If the radius of curvature of the path of two particles of the same mass are in the ratio \( 3:4 \), then in order to have constant centripetal force, their velocities will be in the ratio of:

• A. \( \sqrt{3} : 2 \)
• B. \( 1 : \sqrt{3} \)
• C. \( \sqrt{3} : 1 \)
• D. \( 2 : \sqrt{3} \)

Answer 1

The Correct Option is A

To determine the velocity ratio of two particles with identical mass under constant centripetal force, utilize the centripetal force formula: \(F_c = \frac{mv^2}{r}\). Here, \(m\) denotes particle mass, \(v\) signifies particle velocity, and \(r\) represents the radius of curvature.

Given that the masses are equal and the centripetal force is constant, we can equate the expressions for both particles and derive their ratio:

\(\frac{mv_1^2}{r_1} = \frac{mv_2^2}{r_2}\).

This simplifies to:

\(\frac{v_1^2}{r_1} = \frac{v_2^2}{r_2}\).

Rearranging to establish the relationship between their velocities and radii yields:

\(v_1^2 \cdot r_2 = v_2^2 \cdot r_1\).

With the radii given in the ratio \(r_1 : r_2 = 3:4\), substitution results in:

\(v_1^2 \cdot 4 = v_2^2 \cdot 3\).

Solving for the velocity ratio \(\frac{v_1}{v_2}\):

\(\frac{v_1^2}{v_2^2} = \frac{3}{4}\).

Taking the square root of both sides provides the final velocity ratio:

\(\frac{v_1}{v_2} = \frac{\sqrt{3}}{2}\).

Therefore, the velocities are in the ratio \(\sqrt{3} : 2\).