Problem — Rotating disc struck tangentially

Problem

A uniform disc ($M = 4\;\mathrm{kg}$, $R = 1\;\mathrm{m}$) is free to rotate about its own centre on a frictionless table, initially at rest. A small ball ($m = 0{,}5\;\mathrm{kg}$, $v_0 = 10\;\mathrm{m/s}$) strikes the rim tangentially and sticks to it. Calculate the final angular velocity of the system.

Solution

We choose the centre of the disc as the pole: the pivot’s reaction has zero arm, so angular momentum is conserved. Before the collision only the ball has angular momentum (arm $R$, tangential velocity): $L^A = m\,v_0\,R = 0{,}5\cdot 10\cdot 1 = 5\;\mathrm{kg\,m^2/s}$ After the collision they rotate together, with moment of inertia $I_\text{tot} = \tfrac{1}{2}MR^2 + mR^2$: $I_\text{tot} = \tfrac{1}{2}\cdot 4\cdot 1 + 0{,}5\cdot 1 = 2 + 0{,}5 = 2{,}5\;\mathrm{kg\,m^2}$ Conservation $L^A = I_\text{tot}\,\omega$: $\omega = \frac{5}{2{,}5} = \ev{2\;\mathrm{rad/s}}$

Links

Topics: Rotational dynamics Concepts: Conservation of angular momentum · Inelastic collision · Moment of inertia Skills: Choice of pole Objects: Rotating disc