Question

Derive formula of moment of inertia of hollow sphere ​

Answer 1

Answer:

I = mr2

If we apply differential analysis we get;

dl = r2 dm

We have to find the dm,

dm = dA

Here, A is the total surface area of the shell = 4πR2

dA is the area of the ring formed by differentiation and is expressed as;

dA = R dθ × 2πr

2πr is the circumference of the ring

R dθ is the thickness

Note: We get R dθ from the equation of arc length which is S = R θ

The next step involves relating r with θ.

If we look at the diagram that is given above, we will see that a right angle triangle with angle θ is present.

We get,

sin θ = = r = R sinθ

Now dA becomes:

dA = 2πR2sinθ dθ

If we substitute the equation for dA into dm, we get:

dm = dθ

We will now substitute the equation given above and for r into the equation for dI. We will get;

dm = sin3 θ dθ

Integrating within the limits of 0 to π radians. From one end to another.

We will get;

I = sin3 θ dθ

Now, we need to split the sin3θ into two, as it depicts the case of integral of odd powered trigonometrical functions. We get;

I = sin2 θ sin θ dθ

However, sin2 θ is normally given as sin2 θ = 1- cos2 θ. Now,

I = (1- cos2 θ) sin θ dθ

After this, we use substitution where u = cos θ. We will get;

I = u2 – 1 du

We have to carry out the integration:

I = u2 – 1 du,

Here integral of u2 du = and integral of 1 du = u

If we substitute the values,

I = {[ ]1-1 – [u]1-1

I = {[ (-1)3 -13] – [-1-1]}

I = {[] – [-2]}

I = { +2}

I = {}

I = x

I = MR²

Do follow

mark as brainliest

Answer 2

Answer:

The best way to calculate themoment of inertia (MOI) of a hollow sphere is to calculate the MOI of two solid spheres, and subtract the MOI of the smaller sphere from the MOI of the larger sphere. where r is the perpendicular distance from the axis of rotation, R is the radius of the sphere, and ρ = M/(4πR3/3).