Question

state and prove theorem of perpendicular Axis

Answer 1

The theorem states that “the moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes in its plane and intersecting each other at the point where the perpendicular axis passes though it.”



Explanation: Let OZ be the axis perpendicular to the plane lamina and passing through point O. Let OX and OY be two mutually perpendicular axes in the plane of the lamina and intersecting at point O. If Ix, ly and lz are the moments of inertia of the plane lamina about the axes OX, OY and OZ respectively, then according to the theorem of perpendicular axes, Iz = Ix + ly

Proof: Let, the lamina consists of n number of particles of masses m1, m2, m3, . mn.

and their positions are at the distances r1, r2, ……rn respectively from the origin O.
For simplicity, let a particle of mass m1, is at point P (x1, y1) and its position is denoted by r1.

So, r12  =  x12 +y12

The moment of inertia of the particle of mass m, about OZ axis is,

I1 = m1r12 = m1 (x12 + y12)

Similarly the of inertia of the particle of mass m2 about OZ axes is, I2 = m2 (x22 + y22) Therefore, the moment of inertia of the whole lamina about OZ axis is,

Iz = I1 + I2 + ……..       + In

= m1 (x12 + y12) + m2 (x22 + y22) + …….. + mn (xn2 + yn2)

= (m1x12 +m2x22 +…..+ mnxn2) + (m1y12 +m2y22  +…..+ mnyn2)

So, Iz = Ix + ly [Proved]

Answer 2

Answer:

Statement of the Perpendicular Axis Theorem - The Perpendicular Axis Theorem states that the moment of inertia for any axis that is perpendicular to a plane is equal to the sum of any two perpendicular axes of the body that intersect the first axis.

Consider a planar lamina composed of a large number of particles in the x-y plane as shown in the figure.

Consider a point of mass m in the given point in the figure

From P draw PN and PN' perpendicular to the x and y axes.

Moment of inertia about the x-axis = my².

The moment of inertia of the whole lamella about the x-axis is given by the relation

Ix = ∑my²-----(1)

The moment of inertia of the entire lamina about why the axis is given by

Iy = ∑mx²-------(2)

Similarly, the moment of inertia of the whole slat about the z-axis is given by the relation,

Iz = ∑mr²

But r² = x² + y²

Therefore,

Iz = ∑m (x² + y²)

From equations (1) and (2) we get:

i.e. Iz = ∑mx² + ∑my²

    (or)

Iz = Ix + Iy.

The Perpendicular Axis Theorem helps to calculate the moment of inertia of a body where it is difficult to access one vital axis of the body.

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