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Chapter 9: Problem 44

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) \(\frac{G M}{2 R}\)

Short Answer

Expert verified
The work done to take \(1 \mathrm{~kg}\) mass from earth surface to infinity will be \(\frac{G M}{R}\)

Step by step solution

01

Find Gravitational Potential Energy

The gravitational potential energy at a distance \( r \) from the centre of the Earth is given by \( - \frac{{GMm}}{r} \). Here \( m = 1 \, \text{kg} \), \( r = R \) and the negative sign indicates that the energy is defined as zero at infinity.
02

Calculation of Work Done

The work done in moving an object from the Earth's surface to an infinite distance is simply the change in potential energy, which is \( - \left( - \frac{{GMm}}{r} \right) = \frac{{GMm}}{R} \) because in moving to infinity, final potential energy will be zero.

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