Name: moment of inertia in a solid sphere
Uploaded: 2025-05-22T09:06:05.000Z
Duration: 1 min 10 s
Channel: JoVE

The mass moment of inertia is a measure of a rigid object&#39;s resistance to rotational motion.
For an elemental mass of an object, the differential moment of inertia equals the product of the differential mass and the square of the distance from the object&#39;s rotational axis.
Integrating this expression over the entire mass distribution yields the object&#39;s moment of inertia.
For a constant density, the density term is factored out of the integral, and the integral reduces to a geometrical factor.
The moment of inertia for a solid sphere about its diameter can be estimated by considering a thin elemental disc.
The differential moment of inertia can be expressed as half the product of the differential mass and the square of its radius.
Integrating the expression over the sphere&#39;s diameter gives the moment of inertia for the sphere.
Expressing the disc radius in terms of the sphere&#39;s radius and adjusting the limits of integration, the integral is simplified.
So, the sphere&#39;s moment of inertia depends on its mass and radius.

mass moment of inertia

Mass Moment of Inertia

The mass moment of inertia is a measure of the resistance of a rigid body to angular acceleration. This physical quantity can be evaluated and analyzed in various rotational motions, from everyday objects such as fans and wind turbines to complex machines used in challenging industries.
Consider a differential mass element on an arbitrary object. The differential moment of inertia can be calculated by multiplying the mass of the element by the square of its distance from the object's rotational axis. Integrating this expression over the entire mass distribution yields the total moment of inertia for the object.
<img alt="Rotational inertia formula, I=∫m r²dm, symbol, dynamics calculation, physics education." src="/files/ftp_upload/14352/14352_PT_Equation_1.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia formula, \(I = \int r^2 dm\), mathematical equation, physics study">
The differential mass equals the product of density and differential volume. The equation is simplified when the density term is factored out of the integral for constant density. The moment of inertia reduces to a geometrical factor.
<img alt="Moment of inertia integral formula, \(I = \rho \int_V r^2 dV\), diagram for physics study." src="/files/ftp_upload/14352/14352_PT_Equation_2.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia formula, I=∫r²dV, integral equation diagram, educational physics concept.">
Spherical metal balls are widely used in bearings due to their unique physical and mechanical properties. When it comes to solid spheres, their moment of inertia can be estimated by considering a thin elemental disc located at a fixed distance from its center. For uniform density, the differential mass can be represented as the product of the density and the differential volume of the elemental disc. The differential moment of inertia can be expressed as one-half of the product of the differential mass and the square of its radius.
<img alt="dI=1/2y²dm formula, static equilibrium equation, fundamental concept in physics analysis" src="/files/ftp_upload/14352/14352_PT_Equation_3.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Rotational inertia equation \(dI=\frac{1}{2}y^2dm\); static equilibrium concept formula.">
Integrating the expression over the sphere's diameter gives the moment of inertia of the sphere. The disc radius can be expressed in terms of the sphere's radius, and the limits of integration are adjusted to match the sphere's diameter. The obtained result gives the moment of inertia of the solid sphere about its diameter.
<img alt="Moment of inertia equation, I=2/5MR², formula for solid sphere." src="/files/ftp_upload/14352/14352_PT_Equation_4.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia equation, I=2/5MR^2, for solid sphere; physics formula.">
The moment of inertia depends on the mass and radius of the sphere.
In conclusion, understanding moments of inertia is essential for mechanical engineers to ensure maximum stability and efficiency during all kinds of rotational movement occurring inside their systems.

The mass moment of inertia is a measure of the resistance of a rigid body to angular acceleration. This physical quantity can be evaluated and analyzed in ...

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Moment of Inertia

KEY TERMS AND CONCEPTS

moment of inertia in a solid sphere

The mass moment of inertia describes how much a rigid body resists angular acceleration. It applies to rotating systems such as fans, wind turbines, and many machines used in industry. Engineers use it to judge how easily an object can start or change its spin.

For an arbitrary object, a small mass element contributes to the total moment of inertia based on its mass and its distance from the rotation axis. The farther the mass lies from that axis, the larger its contribution becomes. Adding all of these small contributions across the whole body gives the full moment of inertia.

When the density is constant, the mass of each small region can be written as density times differential volume. That allows the density term to be taken outside the integral. In that form, the moment of inertia depends only on the object’s shape, so it becomes a geometrical factor.

Solid spheres are useful in bearings because of their physical and mechanical properties. For a uniform solid sphere, the moment of inertia can be found by treating the sphere as many thin elemental discs. Each disc has a differential mass based on density and volume, and its differential moment of inertia is one-half the differential mass times the square of its radius.

Integrating these disc contributions across the sphere’s diameter gives the moment of inertia of the sphere about its diameter. The disc radius is written in terms of the sphere’s radius, and the integration limits match the full diameter. The final result shows that the moment of inertia of a solid sphere depends on both its mass and its radius. Understanding this relationship helps mechanical engineers improve stability and efficiency in rotating systems.

Moment of Inertia in a Solid Sphere

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key terms and concepts

Learn how moment of inertia is calculated for rigid bodies and solid spheres, and why mass and radius affect rotation in engineering systems.

Moment of Inertia in a Solid Sphere | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Moment of Inertia in a Solid Sphere

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