Prove that the moment of inertia of a solid sphere of uniform density, when rotating around a diameter,is 2MR2/5, where M is the mass of the sphere and R is the radius
Rigid Body
A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.
Rigid Body Dynamics
Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).
Prove that the moment of inertia of a solid sphere of uniform density, when rotating around a diameter,is 2MR2/5, where M is the mass of the sphere and R is the radius.Hint: integrate the infinitesimal volume element in spherical coordinates.
Let us consider a solid sphere of uniform density rotating about the z-axis.
The sphere’s moment of inertia can be represented as,
Here, dm and z represent the sphere’s small volume element’s mass, and the volume element’s mass distance from the Z-axis, respectively.
From spherical coordinates,
Here, r and θ represent the volume element’s distance from the origin and the angle that r makes with the z-axis, respectively.
Also,
Here, ρ and dV represent the sphere’s mass density, and the volume element’s volume, respectively.
Thus,
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