Consider a solid sphere with mass m which starts from rest at the top of a frictionless fixed spherical shell of radius R, as shown in the figure below. The moment of inertia of the sphere is I = ma?. The best generalized coordinates are 0, ¢, and r. (a) Using the polar coordinate for the mass m, find the Lagrangian. (b) Then, find the equation of motion using Lagrangian multiplier to determine the forces of constraints when the solid sphere flies off the spherical shell and when the friction is insufficient to stop the rolling sphere from slipping. Note that there are two constraints; (1) the center of the sphere follows the surface of the cylinder r = R+a, and (2) the sphere rolls without slipping a(o – 0) = R0. %3D %3D Disk of mass m, radius a, rolling on a cylindrical surface of radius R.

Consider a solid sphere with mass m which starts from rest at the top of a frictionless fixed spherical shell of radius R, as shown in the figure below. The moment of inertia of the sphere is I = ma?. The best generalized coordinates are 0, ¢, and r. (a) Using the polar coordinate for the mass m, find the Lagrangian. (b) Then, find the equation of motion using Lagrangian multiplier to determine the forces of constraints when the solid sphere flies off the spherical shell and when the friction is insufficient to stop the rolling sphere from slipping. Note that there are two constraints; (1) the center of the sphere follows the surface of the cylinder r = R+a, and (2) the sphere rolls without slipping a(o – 0) = R0. %3D %3D Disk of mass m, radius a, rolling on a cylindrical surface of radius R.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

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).

Question

Consider a solid sphere with mass m which starts from rest at the top of a frictionless
fixed spherical shell of radius R, as shown in the figure below. The moment of inertia
of the sphere is I =ma?. The best generalized coordinates are 0, 6, and r. (a) Using
the polar coordinate for the mass m, find the Lagrangian. (b) Then, find the equation
of motion using Lagrangian multiplier to determine the forces of constraints when the
solid sphere flies off the spherical shell and when the friction is insufficient to stop the
rolling sphere from slipping.
Note that there are two constraints; (1) the center of the sphere follows the surface of
the cylinder r= R+a, and (2) the sphere rolls without slipping a(o - 0) = R0.
R
Disk of mass m, radius a, rolling on a
cylindrical surface of radius R.

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