g at the Surface of a Planet

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Imagine a planet twice as dense as the Earth, but with only half its radius. You land on it and step out. Will you feel heavier, or lighter? Twice the density, so stronger gravity? Or half the size, so weaker? Hmm, hold on to your answer. By the end of this video, we will settle it in two lines. And look, finding g on another planet comes up often in JEE gravitation questions. So, let's begin.

0:32Where g=GM/R² comes from

So, let's build the formula together. Take a planet of mass capital M and radius capital R, okay? And let it be spherically symmetric: the same in every direction from its centre, like a uniform ball. Got it? Now, from the last chapter, you know a beautiful fact. Such a sphere pulls a body on its surface, or outside it, as if all its mass were sitting at its centre. Right? So, put a small body on the surface. In Newton's law of gravitation, which distance goes in the denominator? Capital R, exactly! The body is on the surface, so it sits a distance capital R from the centre. Its mass is small m, and its weight is m times small g, the acceleration due to gravity. That weight is just the planet's pull: m g equals capital G capital M m, over capital R squared. Capital G is the gravitational constant. With me? Now cancel small m from both sides. So, small g equals capital G capital M, over capital R squared. Look, small m is gone! So at one place, heavy and light bodies fall with the same acceleration. Clear? One small note. Here we ignore the Earth's rotation. Its effect is small, and we will handle it later in this chapter, okay?

1:50The same g in terms of density

Now, a second form of the same result. Let rho, the Greek letter that looks like a curly p, be the planet's mean density. Mass is volume times density, and the volume of a sphere is four thirds pi capital R cubed. So capital M equals four thirds pi capital R cubed, times rho. Got it? Now put this capital M into our formula for g. Tell me, which power of capital R is left? Let's see. Small g equals capital G over capital R squared, times four thirds pi capital R cubed rho. Capital R cubed over capital R squared leaves just capital R, to the power one. Right? So, small g equals four thirds pi capital G rho capital R, okay? So at the same density, a bigger planet pulls harder at its surface. Interesting, isn't it?

2:43Comparing two planets

Now, the real use of these formulas: comparing planets. Capital G is the same everywhere. So small g is proportional to capital M over capital R squared. That symbol that looks like a stretched alpha means is proportional to. With me? And from the density form, small g is proportional to rho times capital R, the second line. Same physics, two forms. Clear? So, take a new planet. We mark its values with a little dash called prime, like g prime on screen. How do we get g prime from the Earth's g? Take a ratio! g prime over g equals capital M prime over capital M, times capital R over capital R prime, the whole squared. Look carefully: the radius ratio is upside down, because capital R sits in the denominator. And it is squared. Okay? And a topper's habit: given mass and radius, use the first form. Given density and radius, use the second form. Pick the form that fits the data, and the question is half done. Got it?

3:48Standard values for the Earth

Now, three numbers to know by heart. On the Earth's surface, small g is nine point eight metres per second squared; the s to the power minus two on screen means per second squared. Right? The Earth's radius, capital R, is six point four times ten to the power six metres, okay? And its mass, capital M, is six times ten to the power twenty four kilograms. Clear? But do these three numbers agree with our formula? What do you think? Let's check. Capital G is six point six seven times ten to the power minus eleven. Capital G times capital M, over capital R squared, comes to about nine point eight; the wavy equals sign means approximately. They agree! Perfect!

4:36Example 1: g on a planet of one-tenth mass, half radius

Alright, Example one. Read it with me. Here is a planet with one tenth the mass and half the radius of the Earth. Find small g on it. Pause the video, and try it first. Done? Okay, let's check. Mass ratio: capital M prime over capital M is one tenth. And the radius ratio, upside down, as in our formula: capital R over capital R prime is two. With me? Now, the radius is halved, so small g just doubles, right? Nine point eight, times one tenth, times two? No, no, careful! That is the trap. The radius is squared. Half the radius makes g four times bigger, not two times. So g prime equals nine point eight, times one tenth, times two squared. Clear? Nine point eight times one tenth is zero point nine eight, and times four: three point nine two metres per second squared. So, only one tenth of the Earth's mass, but four tenths of its g. Very good!

5:42Example 2: g on a planet twice as dense, half the radius

Now, remember our puzzle from the start? Let's finish it. Here is a planet with twice the Earth's density and half its radius. Compare its surface gravity with the Earth's. Pause the video, and try it. Done? Okay, let's check. Many students write g proportional to capital M over capital R squared, and say: half the radius, so four times g, and twice the density, so eight times! Right? No, no, wait! The mass did not stay the same; this smaller planet holds less mass. We are given density and radius, so use the density form. g prime over g equals rho prime over rho, times capital R prime over capital R, okay? Twice the density, times half the radius: two times one half, which is one. So g prime equals g! Got it? Let's double check with the mass form. Capital M is proportional to rho times capital R cubed. So capital M prime over capital M is two times one eighth: one quarter. With me? Now, g prime over g equals one quarter, times the radius ratio squared, two squared, which is four. One quarter times four: one again. Both forms agree! Right? So, our puzzle is settled. Twice as dense, half the size, and the same surface gravity as the Earth. Not heavier, not lighter, exactly the same! Did you guess it at the start? Perfect!

7:14Quick revision: g = GM/R², density form, comparing planets

So, quick revision. For a spherically symmetric planet, at its surface, small g equals capital G capital M over capital R squared, or four thirds pi capital G rho capital R, okay? To compare planets, use g proportional to capital M over capital R squared, or g proportional to rho times capital R, whichever fits the data. And the traps: the radius is squared, and the mass itself depends on the density and the radius. Now go and try the practice problems on this page. All the best!