Gravitational force due to sphere

In summary, the problem involves a thin hollow spherical shell with a fixed radius and surface mass density. A particle is dropped from infinity towards the center of the shell and the question is what its speed will be when it reaches the center. The known variables are the surface mass density, the distance between the centers of masses, the radius of the shell, and the gravitational constant. The solution involves using the formula for gravitational force, and considering the conservation of energy for the particle at infinity and when it reaches the shell. The tricky part is determining the force when the particle is in the empty sphere.
  • #1
CyberShot
133
2

Homework Statement

Consider a thin hollow fixed spherical shell of radius R and surface mass density rho. A particle initially at rest falls in from infinity. What is its speed when it reaches the center of the shell?

(Assume that a tiny hole has been cut in the shell to let the particle thru.)

Known:

rho, distance between centers of masses, R, G

Homework Equations



F = -GMm/r^2

The Attempt at a Solution

The source of the force acts through the center of the sphere, right? So I know the force, but the problem says the particle falls from infinity (I'm assuming I'd need to use calculus here then)

The gravitational force from the sphere acting on the particle is originating from the sphere's center of mass, and pulling it towards that center radially. The part that tricks me is bringing the test particle in from infinity, so the g-force is pretty much zero at that point in time, and slowly increases at it gets closer. Seems like I require an infinite sum here, but not sure how to realize that.
 
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  • #2
Use conservation of energy. At infinity, the gravitational potential energy is zero, and the particle is in rest, so its KE=0, too. What is the formula for the gravitational potential energy?

When reaching the sphere, and entering into it, getting at a distance r<R from the centre, only that mass attracts the particle that is confined in the sphere of radius r. What is the force in the empty sphere?

ehild
 

1. What is gravitational force due to a sphere?

The gravitational force due to a sphere is the attractive force exerted by a spherical object on another object due to their masses and the distance between them.

2. How is gravitational force due to a sphere calculated?

The gravitational force due to a sphere is calculated using the equation F = G(m1m2)/r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

3. Does the size of the sphere affect the gravitational force?

Yes, the size of the sphere does affect the gravitational force. The larger the sphere, the greater the gravitational force it will exert on other objects.

4. How does the distance between two spheres affect the gravitational force?

The gravitational force between two spheres is inversely proportional to the square of the distance between them. This means that as the distance between the spheres increases, the gravitational force decreases.

5. Can the gravitational force due to a sphere be negative?

No, the gravitational force due to a sphere cannot be negative. It is always an attractive force, meaning it pulls objects towards the sphere, not away from it.

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