Q4. How much external work is needed to separate two point masses of 2 kg and 3 kg slowly from 1 m apart to infinitely far apart? Take G = 6.67 × 10 N m kg.
The potential energy rises from −G × 2 × 3/1, about −4.0 × 10 J, to zero, so the work needed is about 4.0 × 10 J.
Q5. Lifting 1 kg through 100 km from the Earth’s surface: compare the exact energy with mgh. Take g = 9.8 m s and RE = 6400 km.
Exact: ΔU = mgREh/(RE + h) = 9.8 × 6400 × 100 000/6500, about 9.65 × 10 J. mgh = 9.8 × 100 000 = 9.8 × 10 J, about 1.6% too high.
Q6. What is the gravitational potential at the centre of a uniform thin ring of mass M and radius R?
Every bit of the ring is at the same distance R from the centre, and potentials add as numbers, so V = −GM/R. (The field there is zero, because the pulls cancel, but the potential is not zero.)
Q7. Treating the Earth as a uniform solid sphere, find the gravitational potential at its centre. Take g = 9.8 m s and RE = 6.4 × 10 m.
At the centre V = −3GME/(2RE) = −1.5gRE = −1.5 × 9.8 × 6.4 × 10, about −9.41 × 10 J kg, 1.5 times the surface value of Q2.