Moment of Inertia

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Moment of Inertia

We come across many object in our real life where we are able to see rotating object. Thus this object is sometime easy or  some time it is harder to rotate. The measure of  how harder or easier is the object to rotate is explained by moment of inertia. We will discuss the topic in detail.

Consider a rigid body made up of n no of particles as shown in fig A. Let m₁,m_2,m₃,m_n are the mass of each particle and r₁,r_2,r_3,r_n be the perpendicular distance from axis of rotation . The body is rotating about an axis passing through point O.

Figure A

Rotating Disc

              

Thus moment of inertia is defined as sum o f product of mass of each particle and square of distance of the particles from the axis of rotation.

Mathematically, moment of inertia of this body is given by,

if  ∑ m₁,m_2,m₃,m_n = M

∑ r₁²,r₂²_,r₃²_,r_n²  =R²

 I= ∑ MR²

SI unit of moment of inertia is  Kgm²

Dimension =  L²  M¹  T⁰

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Newton's second law

Moment of inertia can be well explained by Newton's second law. Let us understand this.

We know that for any translation motion,

F = ma ----(1)

Similarly, for any rotational motion, 

τ = I ∝  ----(2)

where  τ  is rotational analogous of force 

I = rotational analogous of mass

∝ = rotational analogous of acceleration

Ex (1) : Consider a point mass rotating about a circle of radius r. As the mass is situated at larger distance from axis of rotation as shown in fig C, the moment of inertia increases which reduce the angular acceleration and the object is very difficult to rotate. The moment of inertia in this case is given by

MR²

Figure C

Ex (2) : Suppose if we shift the mass towards the center , the distribution of mass throughout the system changes. As we can see in figure D, The mass is  distributed near the axis of rotation and  at less distance from axis thus, the moment of inertia reduces which increases the angular acceleration and object is east to rotate.

Figure D

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Ex(3) : Consider a rod of length L as shown in fig E, the rod is made up of n numbers of particles of different masses. If the object is set to rotate about one of it's end, then the most of the masses are at large distance from axis of rotation which results in increase in moment of inertia and thus the angular acceleration decreases making it difficult to rotate.

MI = 1/3ML²

Fig E, Rod rotating about one of the end

Ex(4) : Consider a rod of length L as shown in fig F, the rod is made up of n numbers of particles of different masses. If the object is set to rotate about  it's center, then the most of the masses are at small distance from axis of rotation  as compared to eg (3) which results in decrease in moment of inertia and thus the angular acceleration increases making it difficult to rotate.

MI = 1/12 MR²

Fig F, Rod rotating about center

│<<<Kinetic Energy of rotating body││Application Moment of inertia>>>│

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