Worked exercise — projectile against a hinged rod

Problem

A uniform rod ($M = 3\;\mathrm{kg}$, length $L = 2\;\mathrm{m}$) is hinged at its upper end and hangs vertically, at rest. A projectile ($m = 0{,}1\;\mathrm{kg}$, $v_0 = 200\;\mathrm{m/s}$) strikes it horizontally at the other end and becomes embedded in it. Find the angular velocity $\omega$ of the rod immediately after the impact.

Solution

Choice of pole: we take the hinge. The hinge’s reaction force has zero arm and contributes nothing; the weight has an arm but is not impulsive. So the angular momentum about the hinge is conserved. $L^A$ (projectile only, arm $= L$ because it strikes the end): $L^A = m\,v_0\,L = 0{,}1\cdot 200\cdot 2 = 40\;\mathrm{kg\,m^2/s}$ $L^B$ (rod + projectile rotate together), with $I_\text{tot} = \tfrac{1}{3}ML^2 + mL^2$: $I_\text{tot} = \tfrac{1}{3}\cdot 3\cdot 4 + 0{,}1\cdot 4 = 4 + 0{,}4 = 4{,}4\;\mathrm{kg\,m^2}$ Conservation $L^A = L^B = I_\text{tot}\,\omega$: $\omega = \frac{40}{4{,}4} \approx \ev{9{,}1\;\mathrm{rad/s}}$ Energy check: $E_\text{cin}^A = \tfrac{1}{2}\cdot 0{,}1\cdot 200^2 = 2000\;\mathrm{J}$; $E_\text{cin}^B = \tfrac{1}{2}\cdot 4{,}4\cdot 9{,}1^2 \approx 182\;\mathrm{J}$. The collision is strongly inelastic: about $1818\;\mathrm{J}$ went into heat, sound and deformation.

Note: momentum is not conserved (there is an impulsive reaction from the hinge), but angular momentum about the hinge is.

Links

Topics: Rotational dynamics Concepts: Conservation of angular momentum · Inelastic collision Skills: Choice of pole Objects: Rod · Projectile