Constrained mass point subject to elastic force and weight

In summary, the student attempted to find a solution to a system of equations that describe the motion of a mass point on a sphere. They used spherical coordinates and found the Kinetic energy, potential energy, and Lagrangian. They then found the Hamiltonian and found that the total energy is a constant of motion. They drew a phase portrait and noted that the equilibrium points are at the north and south poles, respectively.
  • #1
Ocirne94
8
0
Hi all, is my solution correct? I was rejected because of this...

Homework Statement


Consider a mass point (mass = m) constrained to move on the surface of a sphere (radius = r). The point is subject to its own weight's force and to the elastic force of a spring (elastic constant = k, rest length = 0) which at the other end is fixed to the sphere's north pole.

Write the Lagrangian and the Hamiltonian of the system.
Write the Lagrange's and Hamilton's equations of motion.
Find the constants of motion.
Give a qualitative description of the point's movement

Homework Equations


None given.


The Attempt at a Solution


There are 2 degrees of freedom. I choose spherical coordinates theta and phi [but is this correct? The point can reach the poles, where those coordinates aren't defined anymore].
The kinetic energy is
[itex]T = \frac{m}{2}\cdot (r^2 \dot\theta^{2} + r^{2}sin(\theta)^{2}\dot\phi^{2})[/itex]
The potential energy is
[itex]V = \kappa\cdot r^{2} (1-cos\theta) + mgr(1+cos\theta)[/itex]
The Lagrangian is simply
[itex]L = T-V[/itex]
and, since there isn't any explicit dependence on time, the Hamiltonian is simply
[itex]H = T+V[/itex], but expressed as a function of the momenta [itex]p_\theta[/itex] and [itex]p_\phi[/itex]. I computed it as [itex]p_\theta \cdot \dot\theta + p_\phi \cdot \dot\phi - L[/itex]

[itex]p_\theta = \frac{\partial L}{\partial \dot\theta}=mr^2\dot\theta[/itex]
[itex]p_\phi = \frac{\partial L}{\partial \dot\phi}=\dot\phi r^{2} sin(\theta)^{2}m[/itex]

Then Lagrange's equations are only computations (I hope I haven't mistaken the derivatives), and so are the Hamilton's.

[itex]p_\phi[/itex] is a constant of motion; the total energy (H or E) is, too. There aren't other constants of motion.

Then I have drawn the chart of V and I have used it to trace a qualitative phase portrait, and I have made basic observations on equilibrium points (one, unstable, when the point is at the south pole; one, stable, when it is at the north pole; and a circumference (a parallel) depending on the mass and the elastic constant.

And now?

Thank you in advance
Ocirne
 
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  • #2
Ocirne94 said:
The potential energy is
[itex]V = \kappa\cdot r^{2} (1-cos\theta) + mgr(1+cos\theta)[/itex]
How do you get (1-cos(θ))? And shouldn't there be a factor 1/2 on that term?
 
  • #3
I get it from geometry: the spring's square length is

[itex]r^2(1-cos\theta)^{2} + (sin\theta^{2})[/itex]

which becomes

[itex]r^2 + r^2cos\theta^2-2r^2cos\theta+r^2sin\theta^2[/itex]

the 2 gets simplified with the 1/2 of the elastic potential formula.
This (I forgot to say) setting potential=0 at the south pole of the sphere.
 
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  • #4
Ocirne94 said:
I get it from geometry: the spring's square length is

[itex]r^2(1-cos\theta)^{2} + (sin\theta^{2})[/itex]
Ah yes, of course.
Everything else looks reasonable to me. Maybe more is wanted on the qualitative description. In general, it will oscillate above and below a latitude corresponding to a stable horizontal orbit, yes? Might it be SHM, in terms of a suitable function of time?
 
  • #5




Hello Ocirne, thank you for sharing your solution with us. Your approach seems to be generally correct, however there are a few points that could be improved upon. Firstly, your choice of spherical coordinates is appropriate for this problem since the point is constrained to the surface of a sphere. However, as you mentioned, at the poles the coordinates are not well defined. In this case, it is better to use a different set of coordinates such as latitude and longitude, or spherical coordinates with a different choice of axis. This will ensure that your equations are well-defined throughout the entire motion of the point.

Secondly, your expression for the potential energy seems to be missing a term. The potential energy should also include the energy of the spring, which is given by 1/2*k*(r-r0)^2, where r0 is the rest length of the spring. This term will contribute to the equations of motion and will also affect the equilibrium points of the system.

Thirdly, you are correct in saying that p_phi is a constant of motion, but there is also another constant of motion in this system which is the z-component of the angular momentum, Lz. This can be derived from the symmetry of the problem and will also affect the equations of motion and the equilibrium points.

Finally, your qualitative description of the point's movement is a good start, but you could also mention the effect of changing the mass and the elastic constant on the motion. For example, a higher mass or a stiffer spring will result in a smaller range of motion for the point.

Overall, your solution is on the right track, but there are a few areas that could be improved upon. Keep up the good work!
 

FAQ: Constrained mass point subject to elastic force and weight

1. What is a constrained mass point?

A constrained mass point is a theoretical point that represents the mass of an object in a given system. It is constrained in that it is not allowed to move freely, but is subject to external forces and constraints.

2. What is an elastic force?

An elastic force is a force that is caused by the stretching or compression of a material. It follows Hooke's Law, which states that the force is directly proportional to the displacement of the material.

3. What is the weight of a constrained mass point?

The weight of a constrained mass point is the force exerted by gravity on the object. It is calculated by multiplying the mass of the object by the acceleration due to gravity.

4. How does an elastic force affect a constrained mass point?

An elastic force will cause a constrained mass point to either stretch or compress, depending on the direction of the force. This will in turn affect the position and motion of the point within the system.

5. What are some real-life examples of a constrained mass point subject to elastic force and weight?

Some examples include a spring being compressed or stretched, a rubber band being stretched, or a trampoline being jumped on. In all of these cases, the mass point is subject to both an elastic force and its weight.

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