Question Video: Using Integration to Calculate the Force Due to Pressure from a Fluid on the inside of a Container Mathematics

Video Transcript

A cubical container with side length 20 centimeters is filled with liquid mercury. Find the fluid force exerted on one side of the cubical container, given that the density of mercury is 13,534 kilograms per meter cubed and 𝑔 equals 9.8 meters per second squared. Give your answer to one decimal place.

With any problem of this kind, the key is to set up the integration to be as simple as possible before making any calculations. Let’s start by making a nice, clear diagram of the scenario. So, we have a cubical container of side length 20 centimeters. And let’s convert this into meters so that we’re working entirely in SI units. So, the side length is 0.2 meters. This container is filled to the brim with liquid mercury, which we’ve represented here in magenta. And we’re asked to find the fluid force on one side of the container. Let’s call this 𝐹.

The total force due to pressure on the surface is given by the pressure multiplied by the surface’s area; 𝐹 equals 𝑃𝐴. However, in this case, the pressure will increase with depth from the free surface of the mercury. So, instead, we will need to integrate the pressure function over one side of the cube. There are multiple approaches to an integration like this. One approach is to reduce the 2D integral to a 1D integral. This can be set up easily because although the surface is two-dimensional, the pressure depends only on one dimension, the depth.

Consider a thin horizontal strip of the surface on the side of the cube of height 𝛿𝑦. This strip also has a width of 0.2 meters, but let’s just call it 𝑤 for now for simplicity. Since pressure varies only with depth and the strip is narrow and horizontal, we can assume that the pressure along it is approximately constant. So, the total force on this strip — let’s call this 𝛿𝐹 — is given approximately by the approximate pressure on this strip 𝑃 multiplied by the area 𝛿𝐴, where 𝛿𝐴 equals 𝑤𝛿𝑦.

The pressure 𝑃 is given by the hydrostatic equation 𝑃 equals 𝜌𝑔𝑦, where 𝜌 is the density of the mercury, 𝑔 is the acceleration due to gravity, and 𝑦 is the depth from the free surface of the mercury. Substituting these into our approximation for 𝛿𝐹, we get 𝛿𝐹 is approximately equal to 𝜌𝑔𝑦𝑤𝛿𝑦. In the limit as 𝛿𝑦 tends to zero, this approximation will become exact, giving us d𝐹 equals 𝜌𝑔𝑦𝑤d𝑦.

We have now essentially turned the problem into a one-dimensional problem since 𝜌, 𝑔, and 𝑤 are all constant, and we’re integrating only with respect to the depth, 𝑦. So, the total force 𝐹 will be given by the integral between some lower and upper limits 𝑦 one and 𝑦 two of 𝜌𝑔𝑦𝑤d𝑦. Remember that we have arranged the problems such that 𝑦 increases as we go down. So, the lower limit 𝑦 one will be at the top of the cube and the upper limit 𝑦 two will be at the bottom of the cube. So, what are 𝑦 one and 𝑦 two?

For the lower limit 𝑦 one, consider a typical point at the top of the cube. 𝑦 is related to pressure by the hydrostatic equation 𝑃 equals 𝜌𝑔𝑦. We are assuming that above the cube is a vacuum. So, the pressure at this point is zero. Since 𝜌 and 𝑔 are nonzero constants, 𝑦 at this point must also be zero. So, 𝑦 one is just equal to zero, and 𝑦 two is just 0.2 meters below that. So, 𝑦 two equals 0.2.

Let’s leave the limits out of the integration for now and plug them in at the end. We can also take 𝜌, 𝑔, and 𝑤 outside of the integration, since these are all constants. So, this gives us 𝜌𝑔𝑤 times the integral between 𝑦 one and 𝑦 two of 𝑦d𝑦. Integrating gives us 𝜌𝑔𝑤 times half 𝑦 squared evaluated between 𝑦 one and 𝑦 two. Taking the common factor of a half outside and evaluating between 𝑦 one and 𝑦 two gives us half 𝜌𝑔𝑤 times 𝑦 two squared minus 𝑦 one squared.

Plugging in all our values gives us one-half times 13,534 times 9.8 times 0.2 times 0.2 squared minus zero squared. Plugging all these into our calculator, we get 530.5. And since we were working entirely in SI units, the unit here is simply newtons.