= Rotational inertia measurement with a falling mass
PASCO's <PASPORT Rotary Motion Sensor Experiment Guide>, Experiment 1, specifies a PS-2120A sensor, a rod carrying two equal-radius weights, a three-step drive wheel, a thread and hanging <mass>, a clamp-on guide wheel, a support, a balance and calipers. The thread leaves the drive wheel tangentially. The sensor records <angular velocity> against <time>; a <linear regression> over the accelerating interval supplies <angular acceleration> $\alpha$. The guide measures the drive radius with calipers and repeats the run with the weights removed to subtract the apparatus's own <moment of inertia>. Its tables are blank: it prescribes a measurement and does not report an observed numerical agreement. <PASPORT Rotary Motion Sensor Experiment Guide>{ref}
With hanging <mass> $m_h$ including its hanger, effective drive radius $b$, no thread slip and negligible guide-wheel inertia, <Newton's second law> gives thread <tension> $F=m_h(g-b\alpha)$. The axial <torque> is $bF$. Under negligible <friction>, the inference is therefore
$$
I_{\mathrm{loaded}}=\frac{m_hgb}{\alpha}-m_hb^2.
$$
Repeat the inference for the unloaded assembly and subtract: $I_{\mathrm{weights}}=I_{\mathrm{loaded}}-I_{\mathrm{empty}}$. Omitting the $m_hb^2$ term incorrectly treats thread <tension> as the hanging weight. Comparing with $(M_1+M_2)R^2$ tests the <point mass> approximation; finite-sized weights also have their own <moments of inertia> about their <centres of mass>. <PASPORT Rotary Motion Sensor Experiment Guide>{ref}
Useful proposed checks go beyond that ideal inference. Verify the <rotary encoder>'s angle scale by turning the shaft through a marked full revolution; check the recorded <time> against an independent clock and the balance against a known <mass>. Measure $b$ at the thread's running surface, including thread thickness; avoid overlapping wraps that change $b$. Release without a push, inspect the <angular velocity> record for a straight interval, and stop before the hanger lands or the thread runs out. Repeat with several hanging <masses>, weight radii and rotation directions. A consistent inferred <moment of inertia> across drive <torques> supports the model; curvature, direction dependence or drift indicates a limitation rather than a new <principal moment of inertia>.
For approximately constant resisting <torque> $\tau_f\geq0$ in one rotation direction, an original calibration model is
$$
m_hgb-m_hb^2\alpha=I\alpha+\tau_f.
$$
Fit the left-hand quantity against $\alpha$ across several drives: the slope estimates the axial <moment of inertia> and the intercept estimates $\tau_f$. This calibration is conditional on <friction> remaining constant over the fitted <angular speeds>; a freely slowing run can expose speed dependence. Sensor differentiation noise, guide-wheel <moment of inertia>, thread slip, flexing connectors and uncertain weight radii limit the inference. Removing weights can also change bearing loads, so empty-apparatus subtraction does not automatically cancel <friction>. These are proposed controls and a model extension, not results claimed by PASCO.
A fixed-axis readout measures $\mathbf n^T\mathbf I_O\mathbf n$, not the full <inertia tensor>. In a proposed generalization, repeat about $x$, $y$, $z$ and the three diagonal directions $(\mathbf e_x+\mathbf e_y)/\sqrt2$, $(\mathbf e_x+\mathbf e_z)/\sqrt2$, $(\mathbf e_y+\mathbf e_z)/\sqrt2$, all through the same $O$. For example,
$$
I_{xy}=I_{O,(\mathbf e_x+\mathbf e_y)/\sqrt2}-\frac{I_{xx}+I_{yy}}2.
$$
Six independent scalar readings then determine the six entries of a <symmetric matrix>. Each mounting needs its own apparatus correction, and different reference points require the <tensor parallel-axis theorem> before the readings can be combined.
Back to article page
