Moment Of Inertia Of Sphere Derivation

Moment Of Inertia is the inherent quality of any object that signifies its tendency not to initiate any angular acceleration, or we can say oppose the rotational motion if it is present at rest.

Whenever the word, moment of inertia comes, you should understand immediately that there is one object that is either in a rotational motion around a fixed rotatory axis or an object is at a pure rest state, and it is forced to get into a rotational motion.

Just like Newton said in one of his laws of motion, “An object will only change its state of action if there is an external force applied to it.” Similar is the case of inertia in rotational motion. Inertia to avoid modification in an object’s actual state of action in rotational motion is called the Moment Of Inertia. In this article, the special considerations are deriving the formulas for the moment of inertia of a solid sphere, the moment of inertia of a hollow sphere, and the moment of inertia of a sphere as a whole.

Moment Of Inertia Of A Sphere?

A Sphere is of two types: a hollow sphere and a solid sphere.

• Hollow Sphere: – A hollow sphere is a type of sphere that is vacant from the inside. The center of a hollow sphere doesn’t possess any material. It can be said like a thinned ball whose mass from the interior has been scrapped out. A hollow sphere is sometimes called a shell.
• Solid Sphere: – A solid sphere is a type of sphere that possesses material in the form of its core inside the wall. The center of a solid sphere is the same as the hollow sphere; the only difference is the prominent availability of the material of the sphere made of.

I = mR²

The moment of inertia calculated for a sphere will come out as different; not all spheres have the exact moment of inertia. Similarly, a single sphere can have a different moment of inertia because the moment of inertia is also dependent on the vertical or horizontal axis of rotation. If the axis of rotation of a hollow or solid sphere changes, their respective moment of inertia will also change simultaneously.

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Terms Involved In The Derivation Of Moment Of Inertia For Sphere

1. Volume Of A Sphere: – The volumes of both the spheres are different from each other.

The volume of a hollow sphere is calculated by subtracting the volume of the interior sphere from the volume of the outer sphere.

The volume of the sphere is:

= 4/3 πR³

Hence, the volume of hollow sphere= volume of outer-sphere – volume of inner-sphere

= 4/3 πR³-4/3 πr³

= 4/3 π(R³-r³)

2. Curved Surface Area Of A Sphere: – A curved surface area is the measurement of the area of only the exterior curvature of a sphere. The curved surface area is also the area calculated by the sphere’s perimeter.

CSA of a sphere

= Curved Surface Area of the outer-sphere – Curved Surface Area of the inner-sphere

= 4 πR² – 4 πr²

= 4 π(R²-r²)

Moment Of Inertia Of A Sphere

Moment Of Inertia of a hollow sphere

The axis of rotation passes through the center of mass of the hollow sphere. A moment of inertia of a hollow sphere will be the same as any axis passing through its center.  Ix = Iy = Iz  = I.

The hollow sphere’s mass is M, and the radius is R.

Let’s assume a dm mass on the shell; we will calculate the moment of inertia of this dm mass and then integrate it later to obtain the moment of inertia for the complete sphere.

dI = dMR²

dM = dA

dA = R dθ × 2πR

Thickness = R dθ

Circumference = 2πR

sin θ =  r = R sinθ

We know A = 4R². Therefore, dA = 2πR2sinθ dθ

Substituting dA in the dm,

dm = (Msinθ/2)dθ

Substitung dm and dA in dI,

dI = (MR²sin³θ/2)dθ

Integrating both sides from 0 to π.

Final Formula: – I = 2/3MR²

Moment Of Inertia of solid sphere

We will calculate the moment of inertia of a solid sphere by integrating multiple inertias of the disc.

I = 1/2MR²

dI = 1/2dMR²

dM =⍴dV

V = 4/3 πR³

dV = πR²dx

dM = ⍴πR²dx

Putting dM in dI.

dI =1/2⍴πR4dx

y² = R² – r²

dI = 1/2⍴π(R² – r²)2dx

Integrating both sides, from 0 to R

I = 8/15⍴πR5

Replacing ⍴ = M/V

⍴ = M/4/3 πR³

Substituting ⍴ in I,

Final Formula: – I = 2/5MR²

Conclusion

A sphere is the three-dimensional shape of a circle. You will get two symmetrical hemispheres if you cut a sphere from its center and any random direction. The moment of inertia of a sphere is defined by the mass and the distance at which we determine the moment of inertia because rotational inertia is a property of mass and length. Therefore, a hollow sphere and a solid sphere have their unique moment of inertia. You learned the moment of inertia of a solid sphere, the moment of inertia of a hollow sphere, and ultimately the moment of inertia of a sphere.

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