The Angular Momentum of a Rigid Body Rotating About a Fixed Axis — Question 5

Question 5

A nonzero measurement intercept is not part of rigid-body inertia

A sensor reports axial angular momentum of a rotor as 7 at $\omega=2\,\mathrm{rad/s}$ and 13 at $\omega=4\,\mathrm{rad/s}$, in $\mathrm{kg\,m^2/s}$. Assume a fixed inertia and an additive constant sensor bias.

1. Infer inertia and bias.

2. Predict the reading at $\omega=-3\,\mathrm{rad/s}$ and the true angular momentum then.

Question 5 - Solution

A nonzero measurement intercept is not part of rigid-body inertia

(a) Derivation and calculation

Write the sensor model as $L_{\mathrm{read}}=I\omega+B$. The two measurements give $$7=2I+B,\qquad13=4I+B.$$ Subtracting, $$\boxed{I=\frac{13-7}{4-2}=3.00\ \mathrm{kg\,m^2}},\qquad \boxed{B=7-3(2)=1.00\ \mathrm{kg\,m^2/s}}.$$

(b) Result and interpretation

At the negative angular velocity, $$\boxed{L_z=I\omega=3(-3)=-9.00\ \mathrm{kg\,m^2/s}},$$ $$\boxed{L_{\mathrm{read}}=L_z+B=-9+1=-8.00\ \mathrm{kg\,m^2/s}}.$$ The ratio $L_{\mathrm{read}}/\omega=I+B/\omega$ is speed dependent because of the bias; it is not the physical inertia.

Physical check

The same inertia and bias reproduce both original readings exactly.