Q4. A horizontal turntable of moment of inertia 200 kg m turns freely at 1.0 rad s. A child of mass 40 kg, standing at the centre, walks out to the edge, 2.0 m from the axis. Treat the child as a particle and find the final angular speed after the child stops walking and rotates with the table at the edge.
At the centre the child adds nothing to I; at the edge the child adds mr = 40 × 2 = 160 kg m. So 200 × 1.0 = (200 + 160)ω, and ω = 200/360, about 0.556 rad s.
Q5. In Q4, compare the kinetic energy before and after. Where does the energy go?
Before: ½ × 200 × 1.0 = 100 J. After: ½ × 360 × (5/9), about 55.6 J. About 44 J of kinetic energy is lost, although angular momentum is conserved. The child’s internal forces absorb about 44.4 J of mechanical energy during the controlled outward walk and stop. This is an endpoint energy balance; it does not determine the detailed split into internal energy and heat. Ideal static contact does not justify treating the foot as sliding radially against friction.
Q6. If the Earth shrank to half its radius with no change of mass, keeping its shape, how long would a day be? Treat the Earth as a uniform sphere.
I = 2MR/5 becomes a quarter as large when R is halved. With no external torque, Iω is constant, so ω becomes four times as large and the day a quarter as long: 24/4 = 6 hours.