Parallel and perpendicular axis theorems are two important theorems on moment of inertia. But then, ‘what is moment of inertia?’. In this article, we will learn about ‘parallel axis theorem’, ’perpendicular axis theorem’, ’how to state and prove the theorem of parallel axis?’, ‘how to state and prove the perpendicular axis theorem?’ and some examples and numericals based on both the theorems of moment of inertia.
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We know that inertia is a property that a body already has, to resist the change in its linear state of motion or state of rest. This is a property that measures the mass of the body. However, the moment of inertia which is denoted by (I) is the measure of property or ability of a body to resist its state of rotational motion. It plays the same exact role in rotational motion as mass plays in linear motion and hence we can draw an analogy between them.
There are two theorems on moment of inertia- parallel axis theorem and perpendicular axis theorem which we will discuss them in details in the next topics. So, we can define moment of inertia as the property of a body to oppose any change in its state of uniform motion or its state of rest. I=Mr2, where r is the perpendicular distance of a particle from the rotational axis and M is the mass of the body in rotational motion. (Equation-1)
Also check-
Moment of inertia of a body made of number of particles i.e. in case of discrete distribution of particles (see figure-1) –
(Fig-1)
I=m1r12+m22+m32+… (Equation-2)
For continuous distribution of mass, we consider mass of dm at position r in the particle (see figure-2). (Fig-2)
Then,
dI=dm.r2=>I=r2dm (Equation-3)
NOTE- 1. Moment of inertia is not a vector or a scalar quantity. It is actually a tensor quantity.
2. It is not a constant for a body as it depends on the axis of rotation.
3. The higher is the mass of a body, the larger is its moment of inertia.
4. The farther the mass is distributed from the axis, the larger is the moment of inertia.
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Let us suppose that we want to find the moment of inertia I of a body having mass M about any axis given to us. By equation-3, we can find it using the definition of moment of inertia. But, it is more convenient to find it if we know the moment of inertia of center of mass of the body about an axis parallel to the given axis which passes right through the body’s center of mass. Let h be the perpendicular distance between the axis through center of mass and the given axis (they must be parallel to each other), then rotational inertia I about the given axis is-
I=ICOM+Mh2 (Formula of theorem of parallel axis) (Equation-4)
We can think distance h as the distance by which we have shifted the rotational axis from being passing through its center of mass (see figure-3)
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(Fig-3)
Statement- The Parallel axis theorem states that the moment of inertia of a body (rigid body) about an axis is equal to its moment of inertia about an axis passing through center of mass of the body parallel to given axis plus the product of mass of the body and the square of the perpendicular distance between the two axes parallel to each other.
Equation-4 gives the formula of theorem of parallel axis.
Proof- See figure-4.
(Fig-4)
Let O be the center of mass of any arbitrarily shaped body. We place the origin at this point O. Now, we consider an axis passing through perpendicular to the plane of the body and another axis passing through point P parallel to the first axis. Let the coordinates at P be (a, b).
Let dm be an element of mass with general coordinates (x, y). The moment of inertia of the body about the axis passing through point P is-
I=r2dm
=[(x-a)2+(y-b)2]dm= (x2+y2)dm-2axdm+(a2+b2)dm (Equation-5)
Now, the middle term of the above equation gives coordinates of the center of mass (multiplied by a constant) and thus it is zero (as we have taken its coordinates at origin). As (x2+y2) =R2, where R is the distance from O to dm. Therefore, the term is just ICOM which is the moment of inertia/rotational inertia of the body about an axis through its center of mass. From the fig-4, we can see the last term of the above equation is Mh2, where Mis the total mass of the body. Therefore equation-5 reduces to equation-4.
NCERT Physics Notes:
We can state perpendicular axis theorem as- the moment of inertia of a body about an axis perpendicular to its plane is equal to the sum of its moments about any two mutually perpendicular axes in its plane which intersect each other at the point where the perpendicular axis pass through it.
Therefore, Iz=Ix+Iy (Equation-6)
Proof- Perpendicular axis theorem derivation is quite simple. Consider a particle of mass of Δm located on x-y plane at (x, y)
(Fig-5)
Ix=∆my2, Iy=∆mx2=>Ix+Iy=∆mx2+∆my2=∆mx2+y2=∆mr2
Now, the distance of the particle from z-axis is equal to r.
Iz=∆mr2
Iz=Ix+Iy ( Formula of theorem of perpendicular axis)
Also read -
We use the parallel axis theorem- I1 =Icom+ma2 and I2=Icom+mb2
Therefore, I1 - I2=m(a2-b2)
First we will find moment of inertia about an axis passing through its center and
perpendicular to the plane i.e. Icom
For this, consider an elementary ring of mass dm and width dr at a distance r from the center of
disc. Then, moment of inertia for the elementary ring is- dI=dm.r2
As, dm=MR2.2πrdr
Then, Icom =dIring=0RMR2.2πrdr.r2=MR2/2
Now, to find moment of inertia about axis through its tangent, we use parallel axis theorem-
I=Icom+ MR2, as h=R here.
I=3MR22
We use parallel axis theorem, I=ICOM+Mh2
ICOM=25mR2 , d=R here
Therefore, I=75mR2
We already know that, Icom= MR2/2
By perpendicular axis theorem formula, Iz=Ix+Iy
=> MR2/2= 2ID, as the ring is symmetrical, thus Ix=Iy=ID
=>ID=(14)MR2
Using perpendicular axis theorem-
Ix=Iy
2Ix= 1.6 Ma2
Ix=0.8Ma2
IAB=Ix+M(2a)2=4.8Ma2
Iz=Ix+Iy , where I’s are the moments of inertia in Z, X and Y axes respectively.
I=ICOM+Mh2, h is the perpendicular distance between the axis passing through the COM and the axis about which we have to find the moment of inertia. Both of these axes of rotation must be parallel to each other.
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