# Moment of inertia of spherical shell

• tsw99
In summary, the moment of inertia of a spherical shell of radius R and mass M along its rotation axis is given by 2/3MR^2. However, when attempting to calculate this, the individual moment of inertia of a ring (I=MR^2) is used instead of the moment of inertia of a spherical shell. This results in an incorrect calculation due to the spherical shell being an assembly of rings with varying radii. The correct integrand for calculating the moment of inertia is sin^3θ instead of sin^4θ.
tsw99

## Homework Statement

Moment of inertia of spherical shell of radius R, mass M along its rotation axis is given by $$\frac{2}{3}MR^{2}$$
I am trying to calculate this

## The Attempt at a Solution

This is my attempt but is unsuccessful,
since the spherical shell is an assembly of rings (of varying radius), and the MI of a ring is
$$I=MR^{2}$$
Hence $$dI=y^{2}dm$$
$$I=\int y^2(2\pi \sigma ydz$$
Using $$y=Rsin\theta$$ and $$z=Rcos\theta$$
I get:
$$I=2 \pi \sigma R^{4} \int sin^{4}\theta d\theta =2 \pi \sigma R^{4} \frac{3\pi}{8} =\frac{3\pi MR^{2}}{16}$$
which is incorrect.

Which step I have gone wrong? Thanks

Last edited:
The constant surface charge refers to the spherical surface element which is R^2 sinθ dφ dθ in the spherical polar coordinates. After integrating for φ for a ring, it is dA=2πR^2 dθ. You have to multiply this by σ to get dm, and by the square of the distance from the axis, (Rsinθ)^2. So you have only sin^3 in the integrand. ehild

## 1. What is the formula for calculating the moment of inertia of a spherical shell?

The formula for calculating the moment of inertia of a spherical shell is I = (2/3) * m * r^2, where m is the mass of the shell and r is the radius of the shell.

## 2. How does the moment of inertia of a spherical shell compare to that of a solid sphere?

The moment of inertia of a spherical shell is 2/3 times that of a solid sphere with the same mass and radius. This is because the mass of a solid sphere is distributed evenly throughout its entire volume, whereas a spherical shell only has mass concentrated at its surface.

## 3. Can the moment of inertia of a spherical shell be negative?

No, the moment of inertia of any object cannot be negative as it is a measure of an object's resistance to changes in rotational motion. Negative values would not make physical sense.

## 4. How does the moment of inertia of a spherical shell change with respect to the radius?

The moment of inertia of a spherical shell is directly proportional to the square of the radius. This means that as the radius increases, the moment of inertia also increases, and vice versa.

## 5. How does the moment of inertia of a spherical shell change if the mass is concentrated towards the center instead of the surface?

If the mass of a spherical shell is concentrated towards the center instead of being evenly distributed at the surface, the moment of inertia will decrease. This is because the mass is now closer to the axis of rotation, reducing its moment arm and thus the resistance to rotational motion.

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