Question
26. (b) Prove the theorem of parallel axes.
(Hint: If the centre of mass is chosen to be the origin ∑ miri = 0).
Solution
The theorem of parallel axes states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.
Suppose a rigid body is made up of n particles, having masses m1, m2, m3, … , mn, at perpendicular distances r1,r2, r3, … , rn respectively from the centre of mass O of the rigid body.
The moment of inertia about axis RS passing through the point O,
The moment of Inertia of all the particles about the axis passing through the centre of mass is zero, at the centre of mass.
That is,
Thus the theorem is proved.
Suppose a rigid body is made up of n particles, having masses m1, m2, m3, … , mn, at perpendicular distances r1,r2, r3, … , rn respectively from the centre of mass O of the rigid body.
The moment of inertia about axis RS passing through the point O,
The moment of Inertia of all the particles about the axis passing through the centre of mass is zero, at the centre of mass.
That is,
Thus the theorem is proved.
