State and prove parallel axes theorem.
The parallel axes theorem, also known as Steiner’s theorem, is a result in mathematics that relates the moment of inertia of a rigid body about any axis to its moment of inertia about a parallel axis through its center of mass.
Statement of the theorem:
The moment of inertia I of a rigid body about any axis parallel to an axis through its center of mass is given by:
I = Icm + Md^2
where Icm is the moment of inertia about an axis passing through the center of mass, M is the mass of the body, and d is the distance between the two parallel axes.
Proof:
Let’s consider a rigid body with mass M and center of mass at point O. Let’s assume we know the moment of inertia Icm of this body about an axis passing through point O. Now, let’s consider another axis parallel to the first one, passing through a point A at a distance d from O.
We want to find the moment of inertia I of the body about axis AA’. To do so, we need to express I in terms of Icm.
Let’s divide the body into small elements of mass dm, and let x be the distance of any such element from axis AA’. Then, the moment of inertia of this small element about axis AA’ is given by:
dI = x^2 dm
Integrating over the entire body, we get:
I = ∫x^2 dm
Now, let’s use the fact that the center of mass of the body lies on the line joining O and A, and its distance from axis AA’ is d. Therefore, the distance of any small element of mass dm from axis AA’ can be expressed as:
x = (d + r)
where r is the distance of this element from axis passing through O. Substituting this in the above equation, we get:
I = ∫(d+r)^2 dm
Expanding the square and using the fact that dm = M/N, where N is the total number of small elements, we get:
I = Md^2 + ∫r^2 dm + 2d∫r dm
The second term in this expression is just the moment of inertia Icm of the body about axis passing through O. The third term is zero, since the integral of r dm over the entire body is equal to zero (this follows from the definition of center of mass).
Therefore, we have:
I = Icm + Md^2
This proves the parallel axes theorem.
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