Finding gravitational potential inside solid sphere

In summary, the conversation discusses finding the gravitational potential and force inside and outside a solid sphere (the Earth) of radius R for a mass m. The approach involves treating the Earth as a collection of spherical shells and using integration to derive the potential and force equations. However, there is a discrepancy in the calculation of the constant CM for the potential inside the sphere, which is later resolved by correcting the calculation for the potential due to the shells enclosing the mass m.
  • #1
demonelite123
219
0
So I am given that the gravitational potential of a mass m a distance r away from the center of a spherical shell with mass m' is -Cm'/r for m outside the shell and constant for m inside ths shell.

I am to find the potentials inside and outside a solid sphere (the earth) of radius R as well as the gravitational force inside and outside on a mass m.

I thought of the Earth as a lot of spherical shells of mass dm so if the mass of the solid sphere is M, i integrated for example -Cdm/r from m = 0 to m = M to get -CM/r outside the sphere. Then taking the negative gradient, i find F = (-CM/r2) er. Then since the gravitational force on the surface of the Earth is -mg, i see that -CM/R2) = -mg or CM = mgR2.

now for inside the sphere, i have the potential to be D - CM'/r where D is a constant (due to the shells that enclose the mass m) and M' is the total mass of the shells that do not enclose the mass m. since the sphere has uniform density, we have M'/M = r3/R3 so the potential is D - CMr2/R3. Taking the negative gradient once again, i get 2CMr/R3 and since the force at the surface is -mg, i get 2CM/R2 = -mg or CM = (-1/2)mgR2.

but earlier, i got that the constant CM = mgR2.

why does my constant CM have 2 different values? have i done something wrong?
 
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  • #2
i think i figured it out. i didn't calculate the potential correctly for the case that m was inside the earth. while my -CM'/r was correct, the term D was incorrect. after correctly calculating the potential due to all the shells that enclose the mass m, i do indeed get the same value of CM as i did in the other case.
 

1. How is gravitational potential defined?

Gravitational potential is defined as the amount of work required to move an object from an infinite distance to a specific point in a gravitational field. It is a scalar quantity and is measured in joules per kilogram (J/kg).

2. What is the formula for gravitational potential inside a solid sphere?

The formula for gravitational potential inside a solid sphere is V = -(3Gρr^2)/(5R), where G is the gravitational constant, ρ is the density of the sphere, r is the distance from the center of the sphere, and R is the radius of the sphere.

3. How does the gravitational potential inside a solid sphere vary with distance from the center?

The gravitational potential inside a solid sphere decreases as the distance from the center increases. This is because the mass within a certain distance from the center increases as the distance increases, resulting in a stronger gravitational force.

4. What is the difference between gravitational potential inside a solid sphere and outside a solid sphere?

The main difference is that the gravitational potential inside a solid sphere is negative, while outside it is positive. This is because inside the sphere, the gravitational force is attractive, while outside it is repulsive. Additionally, the formula for gravitational potential is different inside and outside the sphere.

5. Can gravitational potential inside a solid sphere be negative?

Yes, the gravitational potential inside a solid sphere can be negative. This occurs when the distance from the center is less than the radius of the sphere. As the distance increases beyond the radius, the gravitational potential becomes positive.

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