A thin rod of length L and mass m is suspended at its midpoint from a long wire,

Question

A thin rod of length L and mass m is suspended at its midpoint from a long wire, see the

figure. The rod performs simple angular harmonic motion with the period Ta. The moment

of inertia of a thin rod about its perpendicular axis can be shown to be I = mL2=12. Then,

an irregular object is hung from the same wire and its period of oscillations is measured to

be Tb. What is the moment of inertia of this object about its suspension axis?

A thin rod of length L and mass m is suspended at its midpoint from a long wire, see the figure. The rod performs simple angular harmonic motion with the period Ta. The moment of inertia of a thin rod about its perpendicular axis can be shown to be I = mL2=12. Then, an irregular object is hung from the same wire and its period of oscillations is measured to be Tb. What is the moment of inertia of this object about its suspension axis?

Explanation / Answer

the moment of inertia of rod is I1 = (mL^2/12)

the angular frequency of rod is w1 = (2pi/Ta)

let the moment of inertia of irregular object be I2

the angular frequency of irregular object is w2 = (2pi/Tb)

we know that

I1 x w1^2 = I2 x w2^2

or I2 = I1 x (w1/w2)^2

or I2 = I1 x (Tb/Ta)^2

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