Overview

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Two soap bubbles float in a vacuum: one of radius three centimetres, and one of radius four centimetres. They touch, and merge into one bubble, at constant temperature. What is the radius of the new bubble? Seven centimetres? Three plus four? Hmm, it is not that simple, right? Hold on to that thought. First we will see how to use this unit's problem bank, and then we will crack this puzzle together. It is a JEE Advanced level problem, straight from this chapter. So, let's begin.

0:39What is in this chapter

So, first, what is this chapter? It is the problem bank for the whole unit on properties of solids and liquids: mixed solved problems, in three parts. Part A is at JEE Main level, okay? Part B has longer problems, at JEE Advanced level. And Part C is concept checks: the ideas that are often tested in statement form. Short questions, but they need really clear basics. Right? And one rule for the whole chapter: take g equal to ten, unless the problem gives another value. Here g is the acceleration due to gravity, and m s to the power minus two means metres per second squared. Ten keeps the numbers clean.

1:25Part A: a 30-minute test

Clear? Now, Part A, the JEE Main part. When should you attempt it? On the first day of the unit, just to have a look? Not on day one! Only after Chapters one to seven, once the whole unit is done. Why? Because these problems mix every chapter: elasticity, fluids, surface tension and heat, all together. So you need all of them in your hand first. With me? And attempt it as a test. Ten solved problems, thirty minutes, a clock on the table. Cover the solutions, and solve each one yourself first, okay? So tell me. Ten problems in thirty minutes: how much time do you get for each one? Thirty divided by ten: three minutes a problem. This sign, a line with a dot above and below, means divided by. So keep an eye on the clock. Got it?

2:19Part B and Part C

Next, Part B, the JEE Advanced part: six longer problems. Most of them join two ideas together in one problem. For example: two bubbles that merge, a U-tube of water that accelerates, and a body that is heated while it loses heat. Our bubble puzzle is one of them. Remember it? Then Part C: six concept checks, on the ideas that come up often in statement form. Let me give you a taste. Is rubber more elastic than steel? Rubber stretches so much, so rubber, right? No, no, careful! That is the trap. Take the same strain in both: strain is the change in length over the original length. Steel needs a much larger stress for it, and stress is the force per unit area. Okay? In physics, elasticity is measured by the modulus: stress over strain, in the elastic limit. Steel's modulus is much larger, so steel is more elastic. One line, and it checks whether your basics are really clear.

3:25Example 1: force to lift a wire ring off water

Now, let's try one from Part A, and it is an easy one. Here it is: a thin wire ring of radius two centimetres lies flat on water. T, the surface tension of water, is zero point zero seven newton per metre: that is what N m to the power minus one means. Find the extra force, beyond its weight, needed to lift the ring off. Pause the video, and try it first. Done? Okay, let's check. Surface tension is force per unit length: T equals F over L, where F is the force and L is the length of the line it pulls along. With me? So here the length is the circumference of the ring, two pi r, where r is its radius. The force is two pi r times T, right? No, no, wait! That is the trap. The ring lies in the surface, so the surface film touches it along two edges: the inner edge and the outer edge. And the film pulls on both of them. Got it? So the force is two times two pi r T, that is, four pi r T, because there are two edges. Clear? Now the numbers, in metres. The radius r is two centimetres, that is zero point zero two metres. Four pi times zero point zero two times zero point zero seven: about zero point zero one seven six newton. The wavy equals sign means approximately, okay? So, roughly zero point zero one eight newton. Forget one edge, and you get exactly half the answer. Don't fall for it!

4:58Example 2: two soap bubbles merge in a vacuum, new radius

Now, our puzzle from the start. This one is from Part B, and it is a hard one. Two soap bubbles, of radii three centimetres and four centimetres, coalesce, that means merge, in a vacuum, at constant temperature. Find the radius of the new bubble. Pause the video, and try it. Done? Okay, let's check. Many students say: two bubbles join, so the volume is conserved, just like two drops merging, right? Oh no, careful! Drops are liquid, but a bubble holds air, and air can be compressed. Its pressure depends on the radius. Remember, a soap bubble has two surfaces, so the excess pressure inside is four T over r. Here T is the surface tension, not temperature, okay? Now, these bubbles are in a vacuum. So what is the pressure of the air inside? Yes, exactly! The outside pressure is zero, so the pressure inside is just the excess pressure: p equals four T over r. Now multiply p by the volume V, which is four over three, times pi r cubed. The dot means times. One r cancels, and p V equals sixteen pi T over three, times r squared. This sign, like a stretched alpha, means proportional to. So p V is proportional to r squared. Got it? Now Boyle's law: the temperature is constant and no air escapes, so the p V of the new bubble equals the p V of the two old ones added together. Cancel sixteen pi T over three: r one squared plus r two squared equals capital R squared. Here r one and r two, with the small subscripts, are the old radii, and capital R is the new one. So, capital R squared is three squared plus four squared: nine plus sixteen, twenty five. And capital R is five centimetres! Not seven, five. Mystery solved. And see what it means: surface area goes as r squared, so the total surface area is conserved. Three, four, five: easy to remember, isn't it?

7:04Quick revision: Parts A, B, C, count every edge and surface

So, quick revision. Part A is a thirty minute test, after Chapters one to seven. Part B has the longer, JEE Advanced problems, and Part C checks your concepts. In surface tension, count every edge and every surface, and take g as ten unless stated. Now set your clock, open Part A, and begin. You can do this. All the best!