You want to find the moment of inertia of a complicated machine part about an ax

Question

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. You find that it requires a torque of 0.157 N. m to turn this body by 20.0 degree, thus twisting the wire. You now remove this torque and release the body from rest. The body oscillates and you measure 125 oscillations in 265 s. (a) What is the torsion constant of the wire? (b) What is the moment of inertia of this body? (c) Write the equation of motion for theta (t) for this body if the initial angular position was counterclockwise.

Explanation / Answer

The period of a torsional harmonic oscillator is 2 pi sqrt( I / kappa ), where I is the moment of inertia
and kappa is the torsion constant (N m per radian). and I is the moment of inertia in kg m^2.
Since Nm = kg m^2 / s^2, the period is in seconds.

In the case at hand, the period is (265/125) seconds, so
(265/125) s = 2 pi sqrt( I / 0.157 N m )
2.12 s / 2pi = sqrt( I / 0.157 N m)
(0.33284 s)^2 = I / 0.157 N m
I = (0.33284)^2 (0.157) kg m^2
= 0.0173928471 kg m^2

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