The tank in Figure P14.14 is filled with water 2.00 m deep. At the bottom of one side wall is a

step 1 All right, so this question has two parts. The first part, they're asking us to find the force b

The tank in Figure P14.14 is filled with water 2.00 m deep....

Key Concepts

Integration Over a Submerged Surface

Because the pressure on a submerged surface varies with depth, the total force is found by integrating the pressure over the entire area of the surface. This involves summing the infinitesimally small forces (pressure multiplied by an area element dA) across the surface, which results in a net force that is typically not acting through the geometric center.

Center of Pressure

The center of pressure is the point on a submerged surface through which the resultant force can be considered to act. It is determined by calculating the moments produced by the differential forces due to the pressure distribution and is often found by dividing the first moment of the pressure distribution by the total force. This concept is particularly important when determining torques or designing supports for submerged surfaces.

Hydrostatic Pressure Distribution

This concept refers to the way pressure varies with depth in a fluid at rest. The pressure increases linearly with depth according to the relation p = ?gh (ignoring atmospheric pressure when appropriate), where ? is the density of the fluid, g is the acceleration due to gravity, and h is the depth. Understanding this varying pressure is crucial when calculating forces on submerged surfaces.

Torque from Distributed Forces

Torque, or moment, in the context of fluid forces, is the rotational effect produced by the force distribution on a surface. When the fluid exerts a distributed force, the torque about a given point (such as a hinge) is calculated by integrating the product of the local pressure force and its lever arm (the distance from the point of rotation). This approach is used to assess the rotational stability or determine the required strength of support structures.

Transcript

00:01 All right, so this question has two parts.

00:04 The first part, they're asking us to find the force by this water in this box exerted on this hinged surface right here of two meters by one meters.

00:17 We know that force is equal to the pressure times the area.

00:24 And once the pressure exerted by our water, it should just be equal to the density times the area.

00:31 Gravitational exploration times the depth.

00:35 I'll call that h.

00:38 And we'd like to integrate over our surface to find the total force.

00:43 So let's find what our d .f, our infinitesimal force on our infinitesimal areas.

00:51 Well, that is, you can just first of all, write that out as row g .h times our infinitesimal area.

00:58 We're going to integrate over.

01:00 But since our pressure only depends on h, we can rewrite this in terms of row g .h times the width, in this case, 2 meters, we'll say w times dh.

01:21 So we get df equals row gwh dh.

01:31 And now to find out our force, we want to integrate over df, which is just this right here.

01:38 We can pull out the constants.

01:40 So we have row gw and we have our integral and what are our bounds? well, our initial depth is right here, which is this is two meters minus one meters, so our initial depth should be one meters.

02:01 And what's the bottom of our tank? this should be a depth of two meters of water.

02:08 So we have these bounds.

02:10 And now we have our integral, which is just hdh.

02:16 And we can just solve this integral, which is pretty simple.

02:19 We have row gw, and this integral should just be 1 over 2h squared.

02:29 And we can start plugging in numbers.

02:31 Well, our density of water is equal to around 1 ,000, or be more specific, 997 kilograms per meters cubed.

02:47 And then we have g.

02:50 And then we said our width is equal to 2 meters.

02:56 So once we plug all of that in, we have two meters times one over two times we have 8 squared.

03:16 In this case, we have 2 meters squared.

03:17 So we have 4 meters squared minus 1 meters squared.

03:26 So we can see that the 2 me, the 2s cancel out.

03:31 We're left with, we're left here with 3 meters cubed right.

03:39 We can see that our meters cube canceled out here.

03:44 So we have 3 times 997 times 9 .8.

03:50 And if we look at our units, we have kilograms meters per second squared, which is just newton's.

03:59 And when you plug that into the calculator, you should get around 29 .3 .3.

04:10 Newtons.

04:14 And now part b asks us to find out about the torque exerted around our hinge.

04:23 So let's say that's our hinge and that's our hatch.

04:26 Again, we have two meters here and one meters here...

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