Q4. A 3 kg box is dragged at constant speed around a rectangular path, 4 m by 2 m, on a rough floor and returned to its starting point. The friction force is 6 N. Find the total work done by friction, and the total work done by gravity.
The path is 2 × (4 + 2), that is 12 m long, and friction opposes the motion all the way, so it does −6 × 12, that is −72 J. Gravity does zero work, because the box ends at the height it started from. Friction depends on the path; gravity does not.
Q5. A 0.2 kg ball is thrown up from a balcony 5 m above the ground and rises 3 m above the balcony before falling to the ground. Taking the balcony as the reference level, find the ball’s potential energy (a) at its highest point and (b) on the ground. Take g = 9.8 m s−2.
(a) At 3 m above the balcony: 0.2 × 9.8 × 3, that is 5.88 J. (b) The ground is 5 m below the balcony: 0.2 × 9.8 × (−5), that is −9.8 J. A negative potential energy only means the ball is below the reference level.
Q6. A particle’s potential energy is V(x) = 2x2 − 8x joules, with x in metres. Where is the force on it zero, and is that a point of minimum or maximum potential energy?
The force is −dV/dx, that is −(4x − 8), which is zero at x = 2 m. There V is 8 − 16, that is −8 J, and V is larger on either side (for example 0 J at x = 0 and x = 4 m), so it is a minimum of potential energy, and x = 2 m is a point of stable equilibrium.