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What is parallel axis theorem and perpendicular axis theorem? Explain with proof

What is parallel axis theorem and perpendicular axis theorem? Explain with proof

Grade:11

1 Answers

Arun
25750 Points
6 years ago
(i) Parallel axes theorem
 
Statement
 
The moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis through its centre of gravity and the product of the mass of the body and the square of the distance between the two axes.
 
Proof
 
Let us consider a body having its centre of gravity at G as shown in Fig.. The axis XX′ passes through the centre of gravity and is perpendicular to the plane of the body. The axis X1X1′ passes through the point O and is parallel to the axis XX′ . The distance between the two parallel axes is x.
 
Let the body be divided into large number of particles each of mass m . For a particle P at a distance r from O, its moment of inertia about the axis X1OX1′ is equal to m r 2.
 
The moment of inertia of the whole body about the axis X1X1′ is given by,
I = Σ mr2  ???(1)
From the point P, drop a perpendicular PA to the extended OG and join PG.
In the ∆OPA,
OP 2 = OA2 + AP 2
r2 = x2 + 2xh + h2 + AP2  ???(2)
But from ∆ GPA,
GP 2 = GA2 + AP 2
y 2 = h 2 + AP 2  ..(3)
Substituting equation (3) in (2),
r 2 = x 2 + 2xh + y 2   ..(4)
Substituting equation (4) in (1),
I0   = Σ m (x2 + 2xh + y2)
= Σmx2 + Σ2mxh + Σmy2
= Mx2 + My2 + 2xΣmh
Here My2 = IG is the moment of inertia of the body about the line passing through the centre of gravity. The sum of the turning moments of all the particles about the centre of gravity is zero, since the body is balanced about the centre of gravity G.
 
Σ (mg) (h) = 0  (or)  Σ mh = 0 [since g is a constant]
equation (5) becomes, I0= Mx2 + IG
Thus the parallel axes theorem is proved.
 
(ii) Perpendicular axes theorem
 
Statement
 
The moment of inertia of a plane laminar body about an axis perpendicular to the plane is equal to the sum of the moments of inertia about two mutually perpendicular axes in the plane of the lamina such that the three mutually perpendicular axes have a common point of intersection.
Proof
Consider a plane lamina having the axes OX and OY in the plane of the lamina as shown Fig. The axis OZpasses through O and isperpendicular to the plane of the lamina. Let the lamina be divided into a large number of particles, each of mass m. A particle at P at a distance r from O has coordinates (x,y).
 
∴r2 = x2+y2
The moment of inertia of the particleP about the axis OZ = m r2. The moment of inertia of the whole lamina about the axis OZ is
I Z= Σmr2
The moment of inertia of the whole lamina about the axis OX is
Ix =Σ my 2
Similarly, I y= Σ  mx 2
From eqn. (2),   I = Σ mr2 = Σ m(x2+y2)
I = Σmx2+ Σ my2 = Ix+ Iy
 Iz = Ix+ Iy

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