Two bodies, each of mass , are kept fixed with a separation . A particle of mass is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is . The correct statement(s) is (are)

Question

Two bodies, each of mass $M$ , are kept fixed with a separation $2 L$ . A particle of mass $m$ is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is $G$ . The correct statement(s) is (are) [2013]

Smart Achievers · Accepted Answer

(2, 4)

From conservation of mechanical energy,

$\frac{- G M m}{L} - \frac{G M m}{L} + \frac{1}{2} m v^{2} = 0 + 0$

$or, \frac{1}{2} m v^{2} = \frac{2 G M m}{L}$ $∴$ $v = \sqrt{\frac{4 G M}{L}} = 2 \sqrt{\frac{G M}{L}}$

Total energy of mass $' m'$ is conserved as there is no external force involved.

Solution

Q.
Two bodies, each of mass $M$ , are kept fixed with a separation $2 L$ . A particle of mass $m$ is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is $G$ . The correct statement(s) is (are) [2013]

1 The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $4 \sqrt{\frac{G M}{L}}$

2 The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2 \sqrt{\frac{G M}{L}}$

3 The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $\sqrt{\frac{2 G M}{L}}$

4 The energy of the mass $m$ remains constant

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Solution

Q. Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are) [2013]

1 The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 4GML

2 The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 2GML

3 The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 2GML

4 The energy of the mass m remains constant