A torsional pendulum consists of a uniform disc of mass m and radius R attached to a thin rod of torsional constant C. Two particles, each of mass m and speed v, strike the disc symmetrically and stick to it as shown. Find the angular amplitude. Take m = 2 kg, v = 20 m s⁻¹, R = 20 cm and C = 80 N m rad⁻¹.

Question

G1 Physics · Accepted Answer

Each incoming particle moves along a line at perpendicular distance R/2 from the disc centre, so total angular momentum before sticking is L=2[mv(R/2)]=mvR. After sticking at the rim, I=(1/2)mR²+2mR²=(5/2)mR². The rotational kinetic energy immediately after impact is L²/(2I)=mv²/5. At maximum twist this becomes (1/2)Cθₘₐₓ², so θₘₐₓ=√(2mv²/5C). Substituting m=2, v=20 and C=80 gives θₘₐₓ=2 rad.