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

Step 1: Given Data: - Masses \(m_1 = m_2\) - Radius ratio \(\frac{r_1}{r_2} = \frac{3}{4}\)

Step 2: Use the Centripetal Force Formula: - Centripetal force \(F = \frac{mv^2}{r}\). - Since the centripetal force is constant, \(F_1 = F_2\): \[ \frac{m_1 v_1^2}{r_1} = \frac{m_2 v_2^2}{r_2} \]

Step 3: Simplify the Equation: - With \(m_1 = m_2\), we get: \[ \frac{v_1^2}{r_1} = \frac{v_2^2}{r_2} \]

\[ \Rightarrow \frac{v_1}{v_2} = \sqrt{\frac{r_1}{r_2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

So, the correct answer is: \(\sqrt{3} : 2\)