力学 Classical Mechanics · Laboratory experiment

Moment of Inertia using a Torsion Pendulum

Aim: to determine the torsional rigidity $C$ and shear modulus $\eta$ of a suspension wire, measure the moment of inertia of regular and irregular bodies, and verify Steiner's parallel-axis theorem using symmetrical twin sliding masses.

Torsion Apparatus & Optical Counter

Optical Sensor: Beam Active

Drag the inertia disc horizontally to apply a twist angle $\theta_0$ (keep within elastic limit ≤ 180°).

Verification Graph: $T^2$ against $d^2$

Slope $= \frac{8\pi^2 m}{C}$, giving wire rigidity $C$ and modulus $\eta$. Y-intercept gives empty cradle inertia $I_0$.

Micrometer Screw Gauge (Wire Radius $r$)

Observation Table

Specimen on Cradle

Mounted Body

Twin Mass Distance $d$ (cm)

Suspension Wire Properties

Wire Material

Wire Length $L$ (cm)

Wire Gauge / Diameter (mm)

Twist Angle

Initial Angle $\theta_0$ (°)

Theory of Torsional Oscillations

When a circular disc or cradle of moment of inertia $I$ suspended by a wire of length $L$, radius $r$, and shear modulus $\eta$ is rotated through an angle $\theta$, the wire experiences a restoring torsional torque $\tau$:

$$ \tau = -C \theta, \qquad C = \frac{\pi \eta r^4}{2 L} $$

where $C$ is the torsional couple per unit twist (rigidity). The equation of motion is:

$$ I \frac{d^2\theta}{dt^2} + C \theta = 0 \implies \frac{d^2\theta}{dt^2} + \omega^2 \theta = 0 $$

This is simple harmonic motion with time period:

$$ T = 2\pi \sqrt{\frac{I}{C}} \implies T^2 = \frac{4\pi^2}{C} I $$

Parallel-Axis Theorem Verification

Placing two identical symmetrical masses (each of mass $m$) at equal distance $d$ on either side of the axis alters the moment of inertia to:

$$ I = I_0 + 2 (I_{cm} + m d^2) = I_1 + 2 m d^2 $$

Substituting into the period formula:

$$ T^2 = \frac{4\pi^2 I_1}{C} + \left(\frac{8\pi^2 m}{C}\right) d^2 $$

A plot of $T^2$ against $d^2$ is a straight line of slope $S = \frac{8\pi^2 m}{C}$. Once $C$ is obtained from the slope, the wire's shear modulus is determined directly without needing an independent torque rig:

$$ \eta = \frac{16\pi L m}{S \, r^4} $$

Experimental Procedure & Precautions

Measure the wire length $L$ with a meter rod and its radius $r$ with a micrometer screw gauge at several positions to obtain an accurate mean.
Mount the twin cylinder masses at distance $d = 4$ cm from the suspension axis.
Twist the disc through $\sim 60^\circ$ and release smoothly. Record the time for 10 oscillations using the optical laser counter.
Repeat for distances $d = 5, 6, 7, 8, 9, 10$ cm.
Plot $T^2$ vs $d^2$. Fit the linear slope and intercept to determine $C$, $\eta$, and $I_0$.
Mount the test specimens (solid cylinder, hollow cylinder, rectangular bar) and measure their periods to determine their experimental moments of inertia.

Viva Questions

Why must the twist angle not exceed the elastic limit?

Hooke's law for shear stress $\tau = \eta \theta$ holds only within the elastic limit. Beyond this limit, permanent set and plastic deformation occur, destroying isochronism.

Why does the period depend so sensitively on wire radius ($r^4$)?

The torsional rigidity $C$ is proportional to $r^4$ because shear strain increases linearly with radial distance from the wire core, and the moment arm of that shear stress also scales with radius (integral over concentric cylindrical shells).

Does gravity affect the period of a torsion pendulum?

No. In contrast to a simple or physical pendulum, gravity acts parallel to the axis of suspension and exerts zero torque about that axis. The restoring torque is purely elastic.