Gravitation Problem: Force of Sphere on Sheet

In summary, the conversation discusses finding the gravitational force of a uniform sphere of mass M and radius R, located at a height h above an infinite sheet of uniform density \rho_{s}. The suggested approach is to find the force of the sheet on the sphere, which is equal to the force of the sphere on the sheet. One method involves using a gaussian cylinder and calculating g using the assumption that all relevant field lines are in the z direction. However, this yields an answer that looks like 4\piG\rhoz instead of the expected 2\piGM\rho. Another method involves using an equation that relates the surface integral of g·dA over the cylinder to the mass within the cylinder, which may be more
  • #1
awri
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Homework Statement


If a uniform sphere of mass M and radius R, is height h above an infinite sheet of uniform density [tex]\rho[/tex][tex]_{s}[/tex], what is the gravitational force of the sphere, on the sheet.


Homework Equations


F=[tex]\frac{GMm}{R^{2}}[/tex]; [tex]\Phi[/tex]=[tex]\frac{GM}{R}[/tex]; [tex]\nabla[/tex][tex]\bullet[/tex]g=4[tex]\pi[/tex]G[tex]\rho[/tex]; U=m[tex]\Phi[/tex]; F=-[tex]\nabla[/tex]U


The Attempt at a Solution


My professor advised me to find the force of the sheet on the sphere since that force would be equal to the force of the sphere on the sheet. So I drew a gaussian cylinder around my "sheet" and attempted to calculate g by saying that [tex]\nabla[/tex][tex]\bullet[/tex]g = [tex]\frac{dg}{dz}[/tex] since all relavent field lines were in the z direction. All I think I need to know is whether or not that was the correct assumption. I don't think it is because it yeilds an answer that looks like 4[tex]\pi[/tex]G[tex]\rho[/tex]z. And the z is an issue. the answer in the back of the book is 2[tex]\pi[/tex]GM[tex]\rho[/tex]. Any help would be much appreciated.
 
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  • #2
There is another way to use the gaussian cylinder. There's an equation that relates the surface integral of dA over the cylinder, to the mass within the cylinder. I think that will be useful here.
 

1. What is the "gravitation problem: force of sphere on sheet"?

The gravitation problem: force of sphere on sheet is a physics problem that involves calculating the gravitational force exerted by a spherical object on a flat sheet placed at a certain distance from the object. This problem is often used in introductory physics courses to demonstrate the principles of gravitation and force.

2. How do you calculate the force of a sphere on a sheet?

To calculate the force of a sphere on a sheet, you can use the formula F = G * (m1 * m2) / d^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses of the sphere and sheet respectively, and d is the distance between the two objects. Alternatively, you can also use the formula F = (G * M * m) / r^2, where M is the mass of the sphere, m is the mass of the sheet, and r is the distance between their centers of mass.

3. What is the significance of the gravitation problem: force of sphere on sheet?

The gravitation problem: force of sphere on sheet is significant because it helps us understand the fundamental principles of gravitation and force. It also has practical applications in fields such as astronomy and engineering, where the force of gravity between different objects must be taken into account.

4. How does the distance between the sphere and sheet affect the force?

The force between the sphere and sheet is inversely proportional to the square of the distance between them. This means that as the distance increases, the force decreases, and vice versa. This relationship is known as the inverse-square law and is a fundamental principle in physics.

5. Are there any real-life examples of the gravitation problem: force of sphere on sheet?

Yes, there are many real-life examples of the gravitation problem: force of sphere on sheet. One example is the gravitational force between the Earth and the Moon, where the Moon's gravitational force creates tides on Earth's oceans. Another example is the gravitational force between the Sun and planets, which keeps the planets in orbit around the Sun. In engineering, this problem is also applicable in situations where the force of gravity between objects must be considered, such as in the construction of bridges and buildings.

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