If mass of earth is M, radius is R and gravitational...

If mass of earth is $M$, radius is $R$ and gravitational constant is $G$, then work done to take $1 \mathrm{~kg}$ mass from earth surface to infinity will be
(a) $\sqrt{\frac{G M}{2 R}}$
(b) $\frac{G M}{R}$
(c) $\sqrt{\frac{2 G M}{R}}$
(d) $\frac{G M}{2 R}$

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Key Concepts

Escape Energy

Escape energy is the minimum energy needed to move an object from the gravitational field of a massive body to a point at infinity, where the gravitational potential energy is taken to be zero. This energy is determined by calculating the difference between the gravitational potential energy at the surface of the body and the energy at infinity.

Gravitational Potential Energy

Gravitational potential energy represents the energy an object possesses due to its position within a gravitational field. For a mass m in the gravitational field of a larger body with mass M, the potential energy is given by -GMm/r, where r is the distance from the center of the large body. This concept is fundamental when analyzing the energy needed to move a mass within a gravitational field.

Work Done Against Gravity

The work done against gravity to move an object is equal to the change in its gravitational potential energy. When moving an object from one point in a gravitational field to another, the work required is calculated as the difference between the gravitational potential energies at the starting and ending points.

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