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## Main Question or Discussion Point

I want to ask the graph of gravitational force against r and gravitational potential energy against r.

First, about gravitational force against r

The shape of the graph is straight line from origin until the surface of the sphere and then curve (decreasing). I understand about the decreasing part because it follows the inverse - square law. But I don't get the part of the straight line.

I saw the derivation:

F = GMm/r

Final form: F = 4/3 π G ρ m r. From this equation, it is obvious that F is proportional to r so the graph will be straight line.

My questions:

1. Why we substitute M only and not m too?

2. Why can we cross out r

Second, gravitational potential energy against r

To find the shape of the graph when r is less than radius of the sphere, I tried to do the same thing (although I don't understand about it):

U = - GMm/r then subs ρ = M / V where V = 4/3 πr

Thanks

First, about gravitational force against r

The shape of the graph is straight line from origin until the surface of the sphere and then curve (decreasing). I understand about the decreasing part because it follows the inverse - square law. But I don't get the part of the straight line.

I saw the derivation:

F = GMm/r

^{2}then subs ρ = M / V where V = 4/3 πr^{3}Final form: F = 4/3 π G ρ m r. From this equation, it is obvious that F is proportional to r so the graph will be straight line.

My questions:

1. Why we substitute M only and not m too?

2. Why can we cross out r

^{3}with r^{2}so it becomes r? What my brain processes is r^{3}is the cube of radius where r^{2}is the square of distance of a point measured from the center of the sphere. They are not the same so why can be crossed out?Second, gravitational potential energy against r

To find the shape of the graph when r is less than radius of the sphere, I tried to do the same thing (although I don't understand about it):

U = - GMm/r then subs ρ = M / V where V = 4/3 πr

^{3}. After crossing out the r, I got U is proportional to r^{2}but my teacher said it was wrong. Where is my mistake?Thanks