Pressure in a Fluid

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Picture three vessels of different shapes, all with the same base area: one narrow at the top, one straight, one wide at the top like a bucket. Fill all three with water to the same height. The wide one holds much more water, right? So, which base feels the biggest force from the water? Most students say: the wide one, obviously! More water, more weight. Hold that thought; soon you will see why that is wrong. Today: pressure in a fluid. Questions on this come up often in JEE, so let's begin.

0:39Pressure: normal force per unit area

So, what is pressure? Push on a surface, and only the part of the push at right angles to it presses in. That is the normal force, F perpendicular; the small upside down T as a subscript means perpendicular. Pressure P equals F perpendicular over capital A, the area: normal force per unit area. And it is a scalar. Right? Its unit is the pascal, written capital P, small a: one newton per square metre; the power minus two means per square metre. And a bigger unit, the atmosphere, a t m on screen: one point zero one three times ten to the power five pascals, which is seventy six centimetres of mercury. Got it? Now think. Take one point deep inside water that is at rest. Which way does the pressure push there: down, up, or sideways? All of them, equally! In a fluid at rest, the pressure at a point acts equally in all directions. No single direction, which is why pressure is a scalar. With me?

1:42Pressure grows with depth

Alright, now dive into a swimming pool. As you go deeper, what happens to the pressure on your ears? It goes up, yes! More water sits above you. So, for a fluid at rest, P equals P naught, plus rho g h. Rho, the Greek letter, is the density of the liquid, and g is the acceleration due to gravity. Okay? Here, P naught, the P with a small zero as subscript, is the pressure at the free surface, the top of the liquid. And h is the depth below that surface. So rho g h is the extra pressure from the liquid above. Clear? Now look at this formula. Is there any area in it? Any shape of the container? No! Only the depth. So points at the same depth, in the same connected fluid at rest, have the same pressure; the double arrow means so. But careful: same fluid, connected, and at rest. Water on one side and oil on the other? Then the rule fails. Right? One more name. Gauge pressure is P minus P atm, the atmospheric pressure; the small a t m below the P is a subscript. It tells you how far the pressure is above the atmosphere. Got it?

2:55The hydrostatic paradox

Now, back to our three vessels. Force equals pressure times area. So the force on the base is P naught plus rho g h, in brackets, times capital A, the base area. Okay? All three have the same height of water, h, and the same base area. So, which base feels the biggest force? Trick question! No base wins. Same h, same area, so the same force on every base. Surprised? This is the hydrostatic paradox. The force on the base depends only on the depth and the base area, not on the shape, and not on the amount of liquid. So where does the extra weight of water in the wide vessel go? The sloping walls take the difference. Clear?

3:41Barometer and U-tube

So, how do we measure the air's pressure? In a barometer, the atmosphere holds up a column of liquid of height h. So P atm equals rho g h. Okay? For mercury, h is about seventy six centimetres; the wavy equals sign means approximately. Our seventy six centimetres of mercury again. Right? Now, water is much lighter than mercury. So will a water barometer be taller, or shorter? Taller, much taller! Smaller rho needs a bigger h for the same pressure: about ten point three metres. About as tall as a three storey building! Got it? And for U tubes and manometers, one golden rule: equate pressures at the same level, in the same liquid. Same level alone is not enough; it must be the same liquid. We will use this in Example two. With me?

4:34Force on a dam wall

Last idea: a dam, a vertical wall of width w, with water of depth capital H behind it. The water's own pressure, rho g times the depth, grows from zero at the top to rho g capital H at the bottom, so on average it is half of that. Times the area, w times capital H, gives F equals half rho g w capital H squared. Okay? Now, where does this total force act? At half the height, right? No, no, careful! The push is stronger near the bottom, so the force acts lower: at capital H over three above the base. And since the pressure is greatest at the bottom, dams are built thicker there. Clear?

5:14Example 1: absolute and gauge pressure 10 m below a lake

Now, Example one. Find the absolute and the gauge pressure ten metres below the surface of a lake. P naught is one point zero one times ten to the power five pascals, and g is nine point eight metres per second squared. Pause the video, and try it first. Done? Okay, let's check. Gauge pressure first: just rho g h. For water, rho is ten to the power three, that is one thousand kilograms per cubic metre. So, one thousand times nine point eight times ten: nine point eight times ten to the power four pascals. Okay? Now, the absolute pressure. What must we add to the gauge pressure? P naught, yes! The air above the lake presses too. Nine point eight times ten to the power four is zero point nine eight times ten to the power five. Add one point zero one: one point nine nine times ten to the power five pascals. Clear? That is about two atmospheres. So just ten metres of water adds roughly one more atmosphere on top of the air. Nice, right?

6:22Example 2: mercury rise when water is poured in a U-tube

Now a slightly harder one. We have a U tube containing mercury. Water is poured into one arm, to a height of thirteen point six centimetres. By how much does the mercury rise in the other arm? Pause and try. Hint: mercury is thirteen point six times denser than water. Done? Okay, let's check. Golden rule. Take the level where the water meets the mercury, in the water arm. Below it, both arms hold only mercury, the same liquid. So the pressures at that level are equal. Okay? Above that level: thirteen point six centimetres of water on one side, some mercury on the other. Mercury is thirteen point six times denser, so that water balances just one centimetre of mercury. So the two mercury levels differ by one centimetre. With me? So the answer is one centimetre, right? No, no, careful! This is the trap. The arms are equally wide, so the mercury pushed down in one arm comes up in the other. It falls half a centimetre on the water side and rises half a centimetre on the other: a gap of one centimetre. So the mercury rises by zero point five centimetres. Got it?

7:37Quick revision: P = P0 + rho g h, U-tube rule, barometer

So, quick revision. P equals P naught plus rho g h: pressure depends only on depth, not on the shape of the vessel. Same level, in the same connected liquid at rest: same pressure. That is your U tube rule. And the barometer: seventy six centimetres of mercury, or about ten point three metres of water. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!