Overview

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Imagine a tunnel dug straight through the Earth, along a diameter, right through the centre to the other side. You stand at one end and simply drop a ball in. What happens to it? Does it stop at the centre? Does it fly out of the far end? And how long before it comes back to you? Hold that thought. We will settle it in Example two. Gravitation comes up often in JEE, and this chapter, the unit problem bank, is where you test yourself on the whole unit. Today I will show you how to use it, then we will solve two of its problems together. So, let's begin.

0:40Part A: a timed test

So, let's open this chapter. First, Part A: eight solved problems at JEE Main level, mixed from the whole unit. Okay? And the course is very clear: attempt Part A as a thirty minute test, after you finish Chapters one to five. First the chapters, then a clock, like a real test. Right? Now, a quick one for you. Eight problems, thirty minutes. How much time can you give each problem? Thirty over eight; the sign on screen is the division sign. That gives three point seven five. So a little under four minutes a problem, yes, exactly! Read it, spot the idea, solve it, and move on. Clear?

1:26Part B: JEE Advanced level

Next, Part B: five longer problems, at JEE Advanced level. With me? What kind of problems? First, a body dropped into a tunnel through the Earth, yes, our puzzle! And a satellite moving on an ellipse, not a circle. Got it? Then the work needed to take a mass from the centre of a shell out to infinity, and the self-energy of a uniform sphere: the work done to build it, layer by layer, from infinity. Okay? And the extra speed a satellite in a circular orbit needs, to just escape. Each one joins two or three ideas of the unit. Right?

2:04Part C: concept checks

And finally, Part C: five concept checks. These concepts are often tested in statement form: true or false. Clear? Let me give you one from Part C, okay? True or false: a satellite that loses energy to air drag slows down. Sounds obvious, doesn't it? Drag slows things down. So, true or false? Let's check with the energy. In a circular orbit of radius small r, the total energy, capital E, is minus capital G, capital M, small m, over two r. Capital G is the gravitational constant, capital M is the Earth's mass, and small m is the satellite's mass. Drag lowers E, making it more negative, so r must get smaller. The orbit shrinks. With me? Now the kinetic energy, capital K, is G M m over two r, which is exactly minus E, okay? So a smaller r gives a bigger kinetic energy. The satellite speeds up! The statement is false. Drag pulls it into a lower orbit, where it moves faster. Remember this one!

3:13Example 1: height where a body loses 36% of its weight

Alright, here is an easy one from Part A to start with. Read it with me: at what height does a body lose thirty six percent of its weight? Pause the video, and try it. Done? Okay, let's check. The body's mass does not change as it goes up. So its weight, m times small g, falls in exactly the same ratio as small g. Right? Now, small g at height h, written g with a small h below, equals g over, one plus h over R, the whole squared, for a point above the Earth's surface. Capital R is the Earth's radius. That is the same as g times, R over R plus h, squared. Clear? But wait. Many students grab the short formula, g times, one minus two h over R, so two h over R is zero point three six, and h is zero point one eight R, quick and easy. Is that fine? No, no, careful! That is the trap. The short formula works only when h is much smaller than the radius. Here the weight drops by more than a third, so h is not small at all. Use the full formula. Losing thirty six percent means keeping sixty four percent, so, R over R plus h, squared, equals zero point six four. Okay? Take the square root: R over R plus h is zero point eight. The double arrow just means, which gives. Then R plus h is R over zero point eight, that is, one point two five R, got it? So h is zero point two five R, a quarter of the radius. With R equal to six thousand four hundred kilometres, h is six thousand four hundred over four: one thousand six hundred kilometres. See? The short formula would have put you too low. Very good!

5:08Example 2: SHM of a body dropped in a tunnel through Earth

Now our puzzle from the start, as a Part B problem. Read it with me: a tunnel is dug along a diameter of the Earth. Show that a body dropped into it performs simple harmonic motion, and find the period. Pause the video, and try it. Done? Okay, let's check. Treat the Earth as a uniform solid sphere, as the course does, and look at the body at a distance small r from the centre. See the graph: inside the sphere, the field grows in a straight line with r. Okay? So inside, the force is minus capital G, capital M, small m, over R cubed, times r. The minus sign means the force points back towards the centre. And G M over R squared is just small g, so the force is minus m g over R, times r. With me? Now think. At the centre, r is zero, so the force is zero. So the body stops at the centre, right? No, no, wait! Zero force means zero acceleration, not zero speed. The body reaches the centre at its fastest and shoots right past. Look at the force: minus a constant k, equal to m g over R, times r. Force proportional to the distance, always pointing back to the centre: that is simple harmonic motion, or SHM, yes, exactly! For SHM, omega, the Greek letter for angular frequency, is the square root of k over m. Put in k equals m g over R, the m cancels, and omega is the root of g over R, clear? So the period, capital T, is two pi times the root of R over g. The wavy equals sign means approximately. Put in the Earth's standard values of R and small g, and it comes to about eighty four point six minutes. Got it? Look at that number: exactly the period of a satellite skimming the Earth's surface! So, our puzzle: the ball does not stop at the centre. It just reaches the far end, comes back, and is in your hand again after about eighty four point six minutes, if nothing like air slows it down. Beautiful, isn't it?

7:18Quick revision: Parts A, B, C and the key gravitation traps

So, quick revision. Part A as a thirty minute test, Part B at JEE Advanced level, and Part C, the concept checks. The short formula, g times one minus two h over R, only when h is much smaller than R; that is the double less than sign. Otherwise use g over, one plus h over R, squared. Air drag lowers the energy, so the orbit shrinks and the satellite speeds up. And a tunnel through a uniform Earth gives SHM, with a period of about eighty four point six minutes. Now set your thirty minute timer, start Part A, and go. Then try the practice problems on each page. All the best!