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

Answer: (A)

$\sqrt{3} : 2$

Answer 2

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: $\sqrt{3} : 2$