Analytical Mechanics: Proving Equilibrium at Bottom of Bowl

Click For Summary
SUMMARY

The discussion focuses on proving that the equilibrium point for a point particle of mass m confined to a frictionless spherical bowl is at the bottom of the bowl. The analysis utilizes potential energy (p.e.) to establish this equilibrium. It is concluded that while a spherical shape is not strictly necessary for equilibrium, the general form of the bowl must maintain certain curvature properties to ensure stability near the bottom. The concepts are derived from the analytical mechanics textbook by Hand and Finch.

PREREQUISITES
  • Understanding of potential energy in mechanical systems
  • Familiarity with degrees of freedom in physics
  • Knowledge of stability criteria in equilibrium analysis
  • Basic concepts of surfaces of revolution in geometry
NEXT STEPS
  • Study the principles of potential energy in mechanical systems
  • Explore the stability of equilibrium points in various geometrical shapes
  • Research surfaces of revolution and their properties
  • Learn about the mathematical criteria for stability in analytical mechanics
USEFUL FOR

Students of physics, particularly those studying analytical mechanics, as well as educators and anyone interested in the stability of equilibrium points in mechanical systems.

noor
Messages
4
Reaction score
0
Homework Statement
A point particle of mass m is confined to the frictionless surface of a spherical bowl. There are 2 degrees of freedom.
1. Prove that the equilibrium point is the bottom of the bowl. ,
2.Does the bowl need to be exactly spherical for this to be true ?
3. Near the bottom of the bowl what is the most general form possible for the shape of the bowl in order to maintain the stability of the equilibrium point at the bottom?
I found this problem in analytical mechanics book(Hand and Finch) and was trying to solve it.


Homework Equations





The Attempt at a Solution


In part one i used the p.e to find the eq point.
im not sure about part 2&3 please !
 
Physics news on Phys.org
For part 2 you could consider various surfaces of revolution like a cone or a parabolic shape and from there show whether a spherical surface is necessary.
 

Similar threads

Equilibrium - Particle in Bowl, looking for guidance
  • · Replies 1 ·
Replies
1
Views
3K
Proving stable equilibrium: Rotating circular hoop
  • · Replies 6 ·
Replies
6
Views
3K
Bead Sliding on Rotating Rod after Motor is Turned Off
  • · Replies 17 ·
Replies
17
Views
4K
Classical Mechanics: Finding force, equilibrium points, turning points
  • · Replies 6 ·
Replies
6
Views
7K
Engineering Help with a presentation on thermal analysis of an opto-mechanical system
  • · Replies 17 ·
Replies
17
Views
5K
Minimizing the friction of the end of a lever against the top of a moving piston
  • · Replies 21 ·
Replies
21
Views
2K
Graduate Regular vs stable orbits in spherically symmetric potentials
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Understanding the Center of Mass in a Hemispherical Bowl in Equilibrium
  • · Replies 2 ·
Replies
2
Views
3K
What Is a Lagrangian and How Do I Use It in Mechanics Problems?
  • · Replies 3 ·
Replies
3
Views
3K
Points of Analyticity and Singularity
  • · Replies 10 ·
Replies
10
Views
1K