The Angular Momentum of a System of Particles — Question 10

Question 10

An open catcher must account for incoming angular momentum

A stationary catcher at radius $R=0.50\,\mathrm m$ receives particles tangentially at speed $u=4\,\mathrm{m/s}$ counterclockwise. Mass arrives at rate $\dot m=0.20\,\mathrm{kg/s}$ and remains at rest in the catcher. Consider moments about the central fixed axis.

1. Find the angular momentum inflow rate.

2. Find the external support torque required to keep the accumulated material at rest, neglecting other moments.

Question 10 - Solution

An open catcher must account for incoming angular momentum

(a) Derivation and calculation

Each unit of incoming mass carries specific angular momentum $$\ell_z=Ru=.50(4)=2.00\ \mathrm{m^2/s}.$$ Multiplying by the arrival rate, $$\boxed{\dot L_{\mathrm{in}}=\dot m\ell_z=.20(2.00)=+.400\ \mathrm{N\,m}}.$$

(b) Result and interpretation

For the catcher as an open system, $$\frac{dL_{\mathrm{stored}}}{dt}=\tau_{\mathrm{ext}}+\dot L_{\mathrm{in}}.$$ The accumulated material stays at rest, so $$\boxed{\tau_{\mathrm{ext}}=-\dot L_{\mathrm{in}}=-.400\ \mathrm{N\,m}}.$$ The support must remove the incoming angular momentum; a closed-system balance without the flux term would miss this torque.

Physical check

The clockwise support torque exactly balances the counterclockwise angular-momentum inflow.