Suppose we could shrink the earth without changing its mass. At w...

Question

Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

Accepted Answer

Hey, everyone in this problem, we're told to assume that the radius of earth is increased while keeping its mass constant. And we're asked to determine the ratio of the new radius to the old one. If the acceleration due to gravity at the surface is half of its original value, you have four answer choices. Option A, the squared of two divided by one. Option B one half, option C one divided by the square root of three and option D one third. So we're looking for the ratio of the new radius to the old. OK. So that means we're gonna have to figure out both the new and old rapist. All right. So let's write out the variables we have. So we're gonna say that subscription one is gonna represent the original values. OK? The values that we typically see on earth. So R one, that radius is gonna be equal to re the radius of the earth. OK? The max and one it's gonna be equal to me and the acceleration due to gravity. G one is just gonna be the acceleration due to gravity on earth GD. Also the three variables we're kind of told some information about in this problem. If we now look at the new scenario, we need to find out R two and we don't know what this is, but we want to determine this ratio. So we do need to find some information about R two and we're told that the mass M two is kept constant. OK. So M two is just gonna be equal to M one which is just the mass of earth that we're told that the gravity is half of the original value. And so G two is gonna be equal to one half G one which is equal to one half the acceleration due to gravity on earth. Ge. So these are our variables. Now, let's think about what equations we have that relate these three variables together. I recall that the acceleration due to gravity on a planet in G is equal to the gravitational constant capital G multiplied by the mass and divided by R squared or R is the. Now we're interested in a ratio of these, the new radius to old radius. We're gonna go ahead and rearrange this equation to isolate for R. And when we do that can we multiply both sides by R squared divide by little G, we get that R squared is equal to capital G multiplied by M divided by little G and we can take the square root. So we get that the radius R is gonna be equal to the square root. Of capital G multiplied by M divided by little G. All right. So this is the radius equation that we're gonna wanna use. And in order to find the ratio of the new to the old, we're gonna have to do this calculation for both. So let's start with the new radius. So the radius R two is going to be equal to the square root, the gravitational constant capital G multiplied by the mass M two divided by the acceleration due to gravity G two. And similarly, we can do the same for the original radius. Our one which is just equal to the radius of earth. This is gonna be equal to the square root of G multiplied by the mass of earth me divided by the, the acceleration due to gravity ge of the earth. All right. So on the right hand side in green, we have the radius of earth or the original radius. There's no simplifying we can do here. But we are gonna go back to our equation on the left for the new radius. OK? Because we have some information about M two and G two, we can substitute in the relationships we were told and try to simplify this. We're using what we know the mass M two we know is the same as the mass of the earth. OK. The original mass and the final masks are the same, it's kept constant. So we can write the numerator as G multiplied by a. Now the acceleration due to gravity we're told is one half of that on earth. And so in the denominator, we have one half multiplied by GE, they're just substituting in those relationships. Now, we can see that we have something very similar to our R one equation and we have the square root of GME divided by GE, we just have this extra one half in the denominator. You recall that we can separate our square root. OK? If things are being multiplied together one half in the denominator is equivalent to multiplying by two in the numerator. So we can simplify this expression as the square root of two multiplied by the square root of G multiplied by me divided by a little Chey. And now this expression, we have the square root of two multiply by this exact re expression will be compared to expressions. We have the square root of two multiplied by the radius of earth R eight. All right. So we can write that R two is going to be equal to the squared of two multiplied by the radius of earth RV. And now what we wanna do is find that ratio, OK? We have an expression for each of these radii we wanna find now that ratio that we were looking. So the ratio we were looking for was the new radius which is R two divided by the old radius R one. If we substitute in our values, this is gonna be equal to the squared of two multiplied by re divided by re these re terms are gonna divide them. We're left with just the square root of two divided by one. OK? That is the ratio of those radii comparing this to our answer choices. And we can see that this is gonna correspond with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

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Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

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