A solid sphere of mass m and radius R rests on a smooth horizontal surface. A horizontal impulse I is applied at height h=4R/5 above the centre. Which option correctly gives both (i) the minimum time after which the highest point touches the ground and (ii) the centre-of-mass displacement during that time?
Options

Question

A solid sphere of mass m and radius R rests on a smooth horizontal surface. A horizontal impulse I is applied at height h=4R/5 above the centre. Which option correctly gives both (i) the minimum time after which the highest point touches the ground and (ii) the centre-of-mass displacement during that time?

Options

G1 Physics · Accepted Answer

The horizontal impulse I gives centre speed V=I/m and angular speed ω=Ih/I about CM. For a solid sphere I about CM=2mR²/5 and h=4R/5, so ω=2I/(mR). The highest point touches the ground after half a rotation relative to translation; using θ=ωt=π gives t=πRm/(2I). Then displacement s=Vt=πR/2. Hence option B.