Question

Four thin uniform rods each of mass m and length I are arranged to form a square. Find
moment of inertia of the system about an axis passing through its centre and perpendicular,
plane. Find also its moment of inertia about an axis passing through one of iya side

Answer 1

Answer:

A rod of length l and mass m has ml²/12 as moment of inertia about an axis through its center of mass.

Now we take four identical copies of the rod above and form a square frame, whose center of mass lies exactly at the geometric center of the square. How can we then use the moment of inertia of a single rod to calculate the moment of inertia of the entire square frame?

The parallel axis theorem to work out the moment of inertia of a rod of length l with it's centre of mass displaced from the axis of rotation by l/2 then multiply this value by four to get the moment of inertia of the whole square.

The parallel axis theorem is:

I=Icm+md²

Where I is the moment of inertia when the object has been displaced, Icm is the moment of inertia of the object when the axis of rotation passes through the centre of mass and d is the distance it is displaced.

For each rod in your square we have:

Icm=ml²/12.

d=l/2.

I=ml²*(1/12+1/4)=ml²/3.

So multiplying by four gives:

I(square)=4ml²/3.

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