10. Conservation of Energy

(I) By how much does the gravitational potential energy of a 58-kg pole vaulter change if her center of mass rises 4.0 m during the jump?

(II) A sphere of radius r₁ has a concentric spherical cavity of radius r₂ (Fig. 8–42). Assume this spherical shell of thickness r₁ - r₂ is uniform and has a total mass M. Show that the gravitational potential energy of a mass m at a distance r from the center of the shell ( r > r₁ ) is given by

U = - (GmM/r).

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(II) A 66.5-kg hiker starts at an elevation of 1150 m and climbs to the top of a peak 2660 m high.

(a) What is the hiker’s change in potential energy?

(II) Two Earth satellites, A and B, each of mass m = 950 kg , are launched into circular orbits around the Earth’s center. Satellite A orbits at an altitude of 4800 km, and satellite B orbits at an altitude of 12,600 km.

(a) What are the potential energies of the two satellites?