Suppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance 𝓍 ≤ R from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius 𝓇 ≤ 𝓍 there is no net gravitational force from the mass in the spherical shell with 𝓇 > 𝓍. a. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of x, R, m, M, and any necessary constants.

Video transcript

Welcome back, everyone. We are told that a gravitational force can be used as a restoring force or a mass M knot oscillating in a hollow drilled through a spherical masses center or a sphere with mass capital MS in a radius R S. It is proven that an oscillating mass with position X where X is between the inner radius and outer radius experiences a net gravitational force from the spherical mass enclosed by R where R is less than or equal to X. While the mass of the spherical shell at X greater than R does not contribute to the net gravitational force of the oscillating mass. Now, we are told that the density of the sphere is uniform and we are tasked with de driving an expression for the gravitational force on the oscillating mass using these four variables right here. So how are we going to go about using this? Well, Noon's law of gravitation says that the gravitational force is equal to big G times mass one which will just have the mass of the oscillating mass inside times the mass of the inner sphere, not the total sphere. We're just gonna have that mass of of the the empty the hollow right divided by distance from the center of gravity squared. Now we are told that the density is uniform. So what we are able to do is we are able to say that the ratio of the masses right, the mass of the inside sphere times the mass of the sphere in total is equal to the ratio of the volumes here. And what is that? Well, we have that this is equal to four pi X cubed over three, divided by four pi times the outer radius cubed over three. And when you simplify this, all this simplifies down to is X cubed over R S cubed multiplying both sides by this MS term. Here we get that the mass of the inside is equal to the mass of the sphere times X cubed over R S cubed. Let's sub in this value into our equation. And let me just scroll down here just a little bit. We have that our gravitational force is therefore equal to big G times M not over X squared times MS times X cubed over R S cubed. You'll see that the word on the bottom cancels out with the cube on top leaving an X giving us a final answer of G M not MS X over R S cubed, which corresponds to our final answer. Choice of B. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.