A tank with a flat bottom of area A and vertical sides is...

A tank with a flat bottom of area $A$ and vertical sides is filled to a depth $h$ with wath water. The pressure is 1 atm at the top surface. (a) What is the absolute pressure at the bot- tom of the tank? (b) Suppose an object of mass $M$ and density less than the density of water is placed in the tank and floats. No water overflows. What is the resulting increase in pressure at the bottom of the tank? (c) Evalu- ate your results for a backyard swimming pool with depth 1.50 $\mathrm{m}$ and a circular base with diameter 6.00 $\mathrm{m}$ . Two persons with combined mass 150 $\mathrm{kg}$ enter the pool and float quietly there. Find the original absolute pressure and the pressure increase at the bottom of the pool.

Transcript

00:01 In this example, we're going to be working with pressures and absolute pressures.

00:06 Okay, so what we have is a tank that's open to the air.

00:09 Okay, it's a cylindrical tank.

00:12 Okay, and it's got vertical sides.

00:18 Okay.

00:19 And this tank is filled with some, to some height h, with some volume of water.

00:27 Okay, and we know the pressure at the surface of the water is atmospheric pressure, so p initial equals one atmosphere equals 101 .3 kilopascals.

00:43 The tank, the bottom of the tank, has an area a of just some number of meters squared.

00:54 Okay, and now what we're going to do is we're going to take an object of some mass.

00:59 I'll just call that m -sum -o.

01:03 That's a mass of object.

01:09 The density of this object is less than that of water, so it'll float.

01:15 Okay, so i'm going to have p object is less than p water or row water.

01:23 Okay, so we're going to put this object in the water.

01:26 It's going to be submerged to some depth d and displace some amount of water.

01:33 So we're at a slightly higher level.

01:36 Okay, and we're given the condition that no water spills over the tank, so all of the water stays in the tank.

01:42 Okay, and what we want to find first is the absolute pressure at the bottom of the tank before we place the object in the water, and then we want to find the increase in pressure at the bottom of the tank due to this object's immersion.

01:58 Okay, so i know absolute pressure, and i'll call that p -ab.

02:07 Let me go back to black here.

02:11 P -sub -a, that equals the pressure at the surface of the water, plus the pressure due to the weight of the water.

02:23 Okay, so that's going to be, we know that's density of water times g -h.

02:31 Okay, that's the weight of the water per year.

02:33 Unit area.

02:35 Okay, so that's my absolute pressure before we submerge an object.

02:40 Right now we're going to take this object and i know the mass or the weight of my object is going to equal the weight of the displaced water.

02:55 Okay, so that's just going to be density of water times the volume of water displaced, which we can write as a times d.

03:10 Okay, for d is my depth.

03:13 Okay, and now let's solve that for d.

03:15 So i get d equals.

03:17 And i forgot a g in here.

03:19 There should be a g on this time of term as well, but we'll see that cancels out, so we don't need to worry about that.

03:25 But i did forget to include that that we need to include that in that half of the equation.

03:30 So our units match.

03:31 But anyways, d is going to equal the mass of the object divided by the density of water times the area of the tank.

03:42 All right...

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