Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take two soap bubbles, one small and one big, and join them with a thin tube. Which way does the air flow? Many students say: from the big bubble into the small one, until both are the same size. Sounds fair, right? Hold on to your answer. Today: six quick concept checks on solids and liquids. Such one line reasoning questions come up often in JEE, so let's begin.

0:33Check 1: why are small drops round?

So, first a tool. Surface tension, capital T, is also the surface energy per unit area: the work needed to make one unit of new surface. So capital W, the work, equals T times delta A, the new area; delta is the little triangle. Okay? Now, check one. Small drops of liquid are almost perfect little balls. Why do they take a spherical shape? Think about energy! Surface tension makes the surface energy as small as possible. The double arrow means so: least energy, so least surface area. With me? And for a given volume, a sphere has the least surface area of any shape. So drops are spherical. That is the full answer. Clear?

1:20Check 2: a steel needle that floats

Check two. Place a steel needle gently on water, and it floats! Steel, sitting on water. How? Look at the surface, not the needle. The needle is not wetted by the water, so it just presses the surface down, like a weight on a stretched rubber sheet. And the surface film holds it up, by surface tension. Okay? Now, add a little detergent to the water. What happens to the needle? It sinks! Detergents lower the surface tension of water. So the film can no longer hold the needle, and down it goes. Got it?

2:00Check 3: is rubber more elastic than steel?

Check three, a famous one. Rubber stretches so easily and snaps right back. Steel hardly stretches at all. So, is rubber more elastic than steel? The graph on the right will help. Hold on. In physics, elastic has an exact meaning. Look at the graph: stress, the force per unit area, against strain, the fractional change in length. On the straight part, O to A, stress is proportional to strain; that sign on screen means proportional to. That is Hooke's law, and it holds only within the elastic limit, B on the curve; strictly, the straight line ends a little earlier, at A, the proportional limit. Beyond B is the plastic region; D marks the ultimate strength, and E, the fracture. Okay? The constant is the modulus. For stretching, it is Young's modulus Y: stress over strain. That is F over A, over delta L over L: force over area, divided by extension over original length. It is the slope of the straight part, and it is how physics measures elasticity. Got it? Now the numbers. For steel, Y is about two times ten to the power eleven pascals; the wavy equals sign means about, and P a is the pascal, one newton per square metre. For rubber, Y is of the order of ten to the power six to ten to the power seven pascals; the single wavy line means of the order of. Compare the powers of ten. Huge gap, right? So for the same strain, steel needs a far larger stress. It resists being deformed much more strongly. With me? So, no! Steel is more elastic than rubber, in the physics sense. Rubber is only easier to stretch. Don't let the everyday meaning trap you!

3:52Check 4: acceleration at terminal velocity

Check four, from viscosity. Drop a small sphere into a viscous liquid, like glycerine. Its weight, m g, the mass times g, the acceleration due to gravity, pulls it down. Pushing up: F B, the buoyant force, with a small B below, and the viscous drag. On the right is its velocity graph; we will use it soon. Okay? The drag comes from Stokes' law, for slow motion: six pi eta r v. Eta, the Greek letter like an n with a long tail, is the coefficient of viscosity; r is the radius and v the speed. So as the sphere speeds up, the drag grows. Right? Now the graph: velocity against time. It rises, then flattens at v t, the dashed line; the small t below means terminal. There, m g equals F B plus six pi eta r v t, the first formula on the figure. The second gives v t itself: rho is the sphere's density, and sigma here is the liquid's density, not surface tension. Clear? Now the check. The sphere is falling at its terminal velocity. What is its acceleration? Is it g? No, no, careful! It is zero. At terminal velocity, the weight is balanced by the buoyant force and the viscous drag. Net force zero, so acceleration zero: constant velocity, the flat part of the graph. Got it?

5:16Example 1: two soap bubbles joined by a tube, which grows

Now, remember our puzzle from the start? Here it is, as Example one. Two soap bubbles of different sizes are joined by a tube. What happens? Hint: think about the pressure inside each bubble. Pause the video, and try it first. Done? Okay, let's check. The pressure inside a bubble is more than outside; the extra is the excess pressure, delta P, okay? Now, a soap bubble has two surfaces, so delta P equals four T over r, where r is the radius. And in brackets: for a drop, with one surface, it is two T over r. Don't mix them up! Now look at the formula. The radius r sits at the bottom. So which bubble has the larger pressure inside: the small one, or the big one? The small one! Yes, exactly. Smaller r, so larger four T over r. The little bubble pushes harder! And air flows from higher pressure to lower pressure. So air flows from the small bubble into the large one, not the other way! The popular answer to our puzzle was the trap. With me? So the small bubble shrinks, and the large one grows. The small one empties into the big one. Puzzle solved!

6:32Example 2: why a pond freezes from the top and fish survive

Now Example two, a little harder, with two parts. Why does a pond freeze from the top, and why can fish survive the winter? Pause and try. Hint: at what temperature is water densest? Done? Okay, let's check. Usually, colder water is denser, and sinks. So in winter, the coldest water should collect at the bottom and freeze there first, right? No, no, careful! Water is special. From zero to four degrees Celsius, water contracts as it warms; the small circle means degrees. So water is densest at four degrees, and below four, it gets lighter as it cools. Okay? So when the surface cools below four degrees, that water becomes lighter. It stays on top, and there it freezes. So the pond freezes from the top. Clear? Part two. The air above stays below zero, day after day. So why does the water below not freeze as well? Look at the ice itself. Ice is a poor conductor of heat, so the ice layer acts like a blanket, and heat leaks out of the water below only very slowly. Right? So the water below stays liquid, at about four degrees Celsius, and the fish survive the winter. Two parts, two reasons: densest at four degrees, and ice is a poor conductor. Very good!

7:58Quick revision: surface tension, elasticity, viscosity, 4 C

So, quick revision. Surface tension keeps the surface energy least; a soap bubble has excess pressure four T over r, so the small one empties into the big one. Elasticity is the modulus, so steel beats rubber; and at terminal velocity, the acceleration, small a, is zero. Water is densest at four degrees Celsius, and ice conducts poorly, so ponds freeze from the top. Now go and try the practice problems on this page. Give the reason in one line, every time. All the best!