Question

state and prove perpendiculer axis theroem​

Answer 1

Answer:

   Let’s consider an object which is having mass m.

   It consists of small particles which are having masses m1, m2, m3…….respectively.

   The perpendicular distance of each particle from the centre of mass is r1, r2, r3…… (as shown in the figure)

According to the definition of Moment of Inertia, the mass moment of Inertia for the whole object is:

I = m1r12 + m1r22 + m1r32 + …..

As we consider the mass of a body (m) to be concentrated at a point. That point is its centre of mass. If this mass m is situated at a perpendicular distance of r from the centre of mass then Moment of Inertia of the whole object is,

I = ∑ mr2

M.O.I of a 2-dimensional object about an axis passing perpendicularly from it is equal to the sum of the M.O.I of the object about 2 mutually perpendicular axes lying in the plane of the object.

According to the above definition of Perpendicular axis theorem can be written as,

IZZ = IXX + IYY

proof:

Let IZ, IX and IY be moments of Inertia about the X, Y and Z axis respectively.

Moment of Inertia about Z-axis i.e.

IZ = ∫ r2.dA …………. (i)

Here, r2 = x2 + y2

Put this value in the above equation

IZZ = ∫ (x2 + y2) . dA

IZZ = ∫ x2.dA + y2.dA

IZZ = IXX + IYY

Hence proved.