A frictionless table has a mass m on it. The table has a hole in it, where mass m is connected to a string that passes through the hole, attached to a hanging mass M. (a) What is the Lagrangian for this system? Using this Lagrangian, find the equations of motion (a = function of position) for each of the generalized coordinates used. (b) Under what conditions does m make a circular motion? (c) What is the frequency of small oscillations (in r) about this circular motion? Further, assume the oscillations are small: use a Taylor Expansion at the point we choose to use the small angle approximation.)
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 frictionless table has a mass m on it. The table has a hole in it, where mass m is connected to a string that passes through the hole, attached to a hanging mass M.
(a) What is the Lagrangian for this system? Using this Lagrangian, find the equations of motion (a = function of position) for each of the generalized coordinates used.
(b) Under what conditions does m make a circular motion?
(c) What is the frequency of small oscillations (in r) about this circular motion? Further, assume the oscillations are small: use a Taylor Expansion at the point we choose to use the small angle approximation.)
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