Question

Calculate the moment of inertia of a uniform solid sphere of mass of $m$ and radius $R$ about its diameter.

   Calculate the moment of inertia of a uniform solid sphere of mass of $m$ and radius $R$ about its diameter.

Practice Problem in Physics for the JEE Main and Advanced

Practice Problem in Physics for the JEE Main and Advanced

Abhay Kumar 2015 Edition

Chapter 7, Problem 28

Instant Answer

Solved by Expert Ravindra Yadav

01/22/2022

Step 1

Step 1: First, we calculate the mass density of the sphere, which is given by the formula: \[ \rho = \frac{m}{\frac{4}{3}\pi R^3} = \frac{3m}{4\pi R^3} \] Show more…

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Calculate the moment of inertia of a uniform solid sphere of mass of $m$ and radius $R$ about its diameter.

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Key Concepts

Spherical Coordinate Integration

Spherical coordinates are particularly useful for integration over spherical objects due to their symmetry. They allow for the volume element of a sphere to be expressed in terms of the radius, polar angle, and azimuthal angle, which simplifies the calculation of integrals in problems involving spheres.

Symmetry in Rigid Body Dynamics

The symmetry of a rigid body, especially in objects like spheres, can simplify the computation of physical quantities such as the moment of inertia. In symmetric bodies, the distribution of mass is even around the axis of rotation, which often leads to standard formulas and reduces the complexity of the integration process.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a particular axis. It is defined as the integral of the mass elements multiplied by the square of their distances from the axis of rotation, and plays a central role in rotational dynamics.

Uniform Mass Distribution

A uniform mass distribution means that the density of the object is constant throughout its volume. This uniformity simplifies the integration process when calculating the moment of inertia, as it allows for the mass to be expressed as a constant times the volume element.

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