How to solve for the point where a mass falls off a sphere in a magnetic field?

In summary: Thanks for the help!In summary, Goldstein follows a coordinate system centered at the bottom of the hemisphere and uses a Taylor expansion to solve for the second equation. The resulting motion is contained in the xz plane if the initial velocity in y is zero.
  • #1
lylos
79
0

Homework Statement


Consider a smooth hemisphere of radius a placed in the Earth's magnetic field. Place a small point mass on the top of the sphere and provide an initial small displacement as to allow the mass to slide down the sphere. Calculate the point where it falls off the sphere.

This is from chapter 2.4 of Goldstein.

Homework Equations


[tex]L=T-V[/tex]
[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=\frac{\partial L}{\partial q}[/tex]


The Attempt at a Solution


First, I followed Goldstein in using a coordinate axis that is centered at the bast of the hemisphere with z pointing to top of sphere. The resulting motion can be contained in the xz plane if we consider the initial velocity in y to be zero.

[tex]L=\frac{1}{2}m(\dot{x}^2+\dot{z}^2)-mgz+\lambda(\sqrt{x^2+z^2}-a)[/tex]

When I transform this to spherical coordinates, keeping in mind that R is constant, I have:

[tex]L=\frac{1}{2}m(R^2\dot{\theta}^2)-mgRCos(\theta)+\lambda(R-a)[/tex]

Which yields the following equations:

[tex]mR\dot{\theta}^2-mgCos(\theta)+\lambda=0[/tex]
[tex]mR^2\ddot{\theta}=mgRSin(\theta)[/tex]
[tex]R-a=0[/tex]

Goldstein states that you would solve the 2nd then solve the 1st and you can then solve for lambda. I am wondering what trick you must use to solve for the 2nd equation. I feel that the small angle approximation won't work here. Please enlighten?
 
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  • #2
Hmm, good question. The taylor expansion may need to go a bit further for something analytical. If it were me, I'd just go to numerics! Let me see if I can find something about it and post back later. My compy is about to run out of juice.
 
  • #3
We have:
[tex]mR^2\ddot{\theta}=mgRSin(\theta)[/tex]

Which can be rewritten:
[tex]mR^2\frac{d\dot{\theta}}{dt}=mgRSin(\theta)[/tex]

We make the substitution:
[tex]\omega=\dot{\theta}[/tex]

Such that:
[tex]mR^2\frac{d\omega}{dt}=mgRSin(\theta)[/tex]

By chain rule:
[tex]\frac{d\omega}{dt}=\frac{d\omega}{d\theta}\frac{d\theta}{dt}=\frac{d\omega}{d\theta}\omega[/tex]

Finally:
[tex]mR^2\frac{d\omega}{d\theta}\omega=mgRSin(\theta)[/tex]

Upon integrating and replacing [tex]\omega[/tex] with [tex]\dot{\theta}[/tex]
You finally have:
[tex]\dot{\theta}^2=\frac{-2g}{r}Cos(\theta)+\frac{2g}{r}[/tex]

Took me a while to find this trick. I wish I could have figured it out myself without having to resort to someone else's work.
 

1. What is the mass on a sphere problem?

The mass on a sphere problem is a physics problem that involves calculating the gravitational potential energy of a mass placed on the surface of a sphere. It is often used to illustrate the concept of gravitational potential energy and its relationship with distance and mass.

2. How do you solve the mass on a sphere problem?

To solve the mass on a sphere problem, you need to know the mass of the object, the radius of the sphere, and the gravitational constant. Using these values, you can calculate the gravitational potential energy using the formula GPE = (G * m * M) / r, where G is the gravitational constant, m is the mass of the object, M is the mass of the sphere, and r is the distance between the object and the center of the sphere.

3. What is the significance of the mass on a sphere problem?

The mass on a sphere problem helps us understand the relationship between mass and distance in terms of gravitational potential energy. It also demonstrates the concept of gravitational potential energy and how it relates to the gravitational force between two objects.

4. What factors affect the gravitational potential energy in the mass on a sphere problem?

The gravitational potential energy in the mass on a sphere problem is affected by the mass of the object, the mass of the sphere, and the distance between the object and the center of the sphere. The gravitational potential energy increases as the mass of the object or sphere increases, and it decreases as the distance between the object and the center of the sphere increases.

5. How is the mass on a sphere problem related to real-life situations?

The mass on a sphere problem has real-life applications in understanding the gravitational potential energy of objects in orbit around a larger mass, such as planets orbiting the sun. It also helps us understand the stability of objects placed on a curved surface, such as a ball placed on a hill.

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