Physics

Problem

Consider two bodies of masses

m_1

and

m_2

separated by a distance

r

. The gravitational potential energy of the system is

U = -G\dfrac{m_1 m_2}{r}

. a. (i) Obtain the differential

\dfrac{\mathrm dU}{\mathrm dr}

in terms of

U

and

r

. b. (ii) Hence show that the fractional change in gravitational potential energy for a small change

\mathrm dr

in separation is given by

\dfrac{\mathrm dU}{U} = -\dfrac{\mathrm dr}{r}

. c. (iii) A spacecraft gradually increases its orbital radius from

r

r+\Delta r

, where

\Delta r = 0.01\,r

. Using the above result, estimate the percentage change in its gravitational potential energy, clearly stating any approximation used.