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:
- \( \sqrt{3} : 2 \)
- \( 1 : \sqrt{3} \)
- \( \sqrt{3} : 1 \)
- \( 2 : \sqrt{3} \)
The Correct Option is A
Approach Solution - 1
To find the ratio of velocities of the two particles having the same mass, we need to consider the condition of constant centripetal force. The formula for centripetal force (\(F_c\)) is given by:
\(F_c = \frac{mv^2}{r}\)
where:
- \(m\) is the mass of the particle,
- \(v\) is the velocity of the particle,
- \(r\) is the radius of curvature.
Since the mass of the two particles is the same and the centripetal force is constant, we can equate the expressions for the two particles and consider their ratio:
\(\frac{mv_1^2}{r_1} = \frac{mv_2^2}{r_2}\).
Simplifying, we get:
\(\frac{v_1^2}{r_1} = \frac{v_2^2}{r_2}\).
Rearranging gives the relation between their velocities and radii:
\(v_1^2 \cdot r_2 = v_2^2 \cdot r_1\).
We know the radii are in the ratio \(r_1 : r_2 = 3:4\). Substituting these values, we have:
\(v_1^2 \cdot 4 = v_2^2 \cdot 3\).
Solve for \(\frac{v_1}{v_2}\):
\(\frac{v_1^2}{v_2^2} = \frac{3}{4}\).
Taking the square root of both sides, we get:
\(\frac{v_1}{v_2} = \frac{\sqrt{3}}{2}\).
Hence, the velocities are in the ratio \(\sqrt{3} : 2\).
Approach Solution -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: \(\sqrt{3} : 2\)
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