Question
(a) Prove the theorem of perpendicular axes.
(Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is (x2 + y2).
Solution
The theorem of perpendicular axes states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.
A physical body with centre O and a point mass m,in the x–yplane at (x, y) is shown in the following figure below.
Moment of inertia about z-axis, Iz = m(x2 + y2)1/2
Therefore,
Ix + Iy = mx2 + my2
= m(x2 + y2)
=
Ix + Iy = Iz
Hence, the theorem is proved.
A physical body with centre O and a point mass m,in the x–yplane at (x, y) is shown in the following figure below.
Moment of inertia about x-axis, Ix = mx2
Moment of inertia about y-axis, Iy = my2 Moment of inertia about z-axis, Iz = m(x2 + y2)1/2
Therefore,
Ix + Iy = mx2 + my2
= m(x2 + y2)
=
Ix + Iy = Iz
Hence, the theorem is proved.
