A thin rod of mass M, length L with the left end at the origin has a density function λ= Ao [1 - (x²/ L²)] and rest on the positive x-axis. Relate M with density constant λo. Find the moment of inertia in terms of M and L only. Find the center of mass in terms of L only. Find the angular velocity w₂ if this rod swings to vertical (pivoted at the origin). The bottom edge of the rod (now vertically hanging) is attached to a spring of constant k and set up as an oscillator. The spring is unstretched when the rod is vertical. Relate the torque on the rod with the angular acceleration. Then find the angular frequency of the oscillation.
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).
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