JEE Mains · Physics · STD 11 - 7. gravitation
A metal wire of uniform mass density having length \(L\) and mass \(M\) is bent to form a semicircular arc and a particle of mass \(m\) is placed at the centre of the arc. The gravitational force on the particle by the wire is _______.
- A \(\frac{\mathrm{GMm} \pi}{2 \mathrm{~L}^2}\)
- B \(0\)
- C \(\frac{\mathrm{GmM} \pi^2}{\mathrm{~L}^2}\)
- D \(\frac{2 \mathrm{GmM} \pi}{\mathrm{L}^2}\)
Answer & Solution
Correct Answer
(D) \(\frac{2 \mathrm{GmM} \pi}{\mathrm{L}^2}\)
Step-by-step Solution
Detailed explanation
\(R = \frac{L}{\pi}\) \(F = \int_{-\pi/2}^{\pi/2} \frac{G m (\frac{M}{L} R d\theta)}{R^2} \cos\theta\) \(F = \frac{GmM}{LR} [\sin\theta]_{-\pi/2}^{\pi/2}\) \(F = \frac{GmM}{LR} (1 - (-1)) = \frac{2GmM}{LR}\) \(F = \frac{2GmM}{L(L/\pi)} = \frac{2GmM\pi}{L^2}\)
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