A non-uniform disk of mass M and radius R has its mass distributed in such a way that the mass per unit area is a function of the radial distance r from the center of the disk: σ(r) = b r , where b is a constant to be determined. What is the rotational inertia of this disk about an axis through the center of mass and perpendicular to the plane of the disk? The area differential can be written as a ring of radius r and thickness dr: dA = 2πrdr.
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).
A non-uniform disk of mass M and radius R has its mass distributed in such a way that the mass per unit area is a function of the radial distance r from the center of the disk:
σ(r) = b r ,
where b is a constant to be determined. What is the rotational inertia of this disk about an axis through the center of mass and perpendicular to the plane of the disk? The area differential can be written as a ring of
radius r and thickness dr: dA = 2πrdr.
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