
State and derive theorem for parallel and perpendicular axes.
Answer
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Hint
According to the perpendicular axis theorem for an axis perpendicular to the plane, the moment of inertia is equal to the sum of the moments of inertia about 2 mutually perpendicular axes in its own plane. And according to the parallel axis theorem, the moment of inertia about any axis is equal to sum of parallel and passing through the center of mass and product of mass of body and square of distance between the axes.
Complete step by step answer
The parallel axis theorem states that, the moment of inertia of a body about any axis is equal to the moment of inertia about parallel axis through its center of mass plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.
Let us consider is the moment of inertia of a body of mass about the axis AB. Let us consider another parallel axis CD passing through the center of mass of the body and at a distance from AB. We consider the moment of inertia about the axis CD of the body as .
Let us consider a particle of mass which is at a distance from the axis CD and at a distance from the axis AB.
So the moment of inertia of this particle about the axis AB is
So to get the moment of inertia about the whole body we use summation. Therefore, we get
Now we can break the whole square in the formula as,
So on breaking the individual terms under the summation we get
Now we can write
Since the value of is constant we can bring it outside summation. So we get
where is the mass of the whole body.
For the third term, comes outside the bracket and since so the third term vanishes.
Therefore, substituting the values we get
.
This is the parallel axis theorem.
The perpendicular axis theorem states that the moment of inertia of a plane lamina about a perpendicular axis is equal to the sum of the moments of inertia of the lamina about any 2 mutually perpendicular axes in its own plane and intersecting each other at the point where the perpendicular axis passes through the lamina.
Let us consider another plane lamina as in the figure.
We consider a particle of mass at the distance , and from the X- axis, Y-axis and Z-axis respectively.
Therefore from the figure we can write,
So the moment of inertia of the particle about X-axis is
Therefore, for the whole bode, we consider summation and get moment of inertia as,,
Similarly moment of inertia of the whole body about Y-axis is
And the moment of inertia about Z-axis is
We can substitute and get
On opening the bracket we get
We can write this as,
This is the perpendicular axes theorem.
Note
The parallel and perpendicular axes theorem are used to find the moment of inertia of various rigid structures about different axes like a rod, disc, sphere, ring, etc. It gives us a simplified way of finding the moment of inertia.
According to the perpendicular axis theorem for an axis perpendicular to the plane, the moment of inertia is equal to the sum of the moments of inertia about 2 mutually perpendicular axes in its own plane. And according to the parallel axis theorem, the moment of inertia about any axis is equal to sum of parallel and passing through the center of mass and product of mass of body and square of distance between the axes.
Complete step by step answer
The parallel axis theorem states that, the moment of inertia of a body about any axis is equal to the moment of inertia about parallel axis through its center of mass plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.
Let us consider

Let us consider a particle of mass
So the moment of inertia of this particle about the axis AB is
So to get the moment of inertia about the whole body we use summation. Therefore, we get
Now we can break the whole square in the formula as,
So on breaking the individual terms under the summation we get
Now we can write
Since the value of
For the third term,
Therefore, substituting the values we get
This is the parallel axis theorem.
The perpendicular axis theorem states that the moment of inertia of a plane lamina about a perpendicular axis is equal to the sum of the moments of inertia of the lamina about any 2 mutually perpendicular axes in its own plane and intersecting each other at the point where the perpendicular axis passes through the lamina.
Let us consider another plane lamina as in the figure.

We consider a particle of mass
Therefore from the figure we can write,
So the moment of inertia of the particle about X-axis is
Therefore, for the whole bode, we consider summation and get moment of inertia as,,
Similarly moment of inertia of the whole body about Y-axis is
And the moment of inertia about Z-axis is
We can substitute
On opening the bracket we get
We can write this as,
This is the perpendicular axes theorem.
Note
The parallel and perpendicular axes theorem are used to find the moment of inertia of various rigid structures about different axes like a rod, disc, sphere, ring, etc. It gives us a simplified way of finding the moment of inertia.
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