Calculate PSI of Hollow Cube with Hole Opened

[daviddjh]

Suppose you have a hollow cube with a volume of 504c³. The inside of the cube has a psi of 94 and is just air. Now suppose that a hole opened in the bottom of this cube with an area of 12.56cm² that leads to a hollow cylinder that is 14 cm long with a radius of 2cm. Now how would you go about calculating the psi of this whole structure after the hole is opened, the air is released and the volume has changed.

 

[Nugatory]

Does the cylinder contain air at normal atmospheric pressure, or is it vacuum?

 

[daviddjh]

Does the cylinder contain air at normal atmospheric pressure, or is it vacuum?

The cylinder would have a vacuum inside it

 

[Nugatory]

The cylinder would have a vacuum inside it

OK - that makes it easy. Boyle's Law, $PV=c$ where $c$ is a constant, will get you an answer in short order.

(Boyle's Law would still get you there if the cylinder weren't a vacuum, but it would be more work).

 

[PJC01]

To calculate the psi of the hollow cube with the hole opened, we first need to determine the total volume of the cube and the hollow cylinder. The total volume can be calculated by subtracting the volume of the hole and the hollow cylinder from the original volume of the cube.

Volume of cube = 504cm^3
Volume of hole = 12.56cm^2 x 14cm = 175.84cm^3
Volume of hollow cylinder = πr^2h = 3.14 x 2^2 x 14 = 175.84cm^3

Total volume = 504cm^3 - 175.84cm^3 - 175.84cm^3 = 152.32cm^3

Next, we need to calculate the new psi of the air inside the structure. Since the air is now contained in both the cube and the hollow cylinder, we can use the combined volume to calculate the new psi.

Combined volume = 152.32cm^3
New psi = (94 x 504cm^3) / 152.32cm^3 = 310.2 psi

Therefore, the psi of the whole structure after the hole is opened and the air is released is 310.2 psi. This is a significant increase from the original psi of 94, as the volume has decreased and the air is now contained in a smaller space. This calculation can be useful in understanding the pressure changes that occur in a confined space when a hole is opened.

 

[PJC01]

To calculate the psi of the entire structure after the hole is opened, we first need to determine the new volume of the structure. We can do this by subtracting the volume of the cylinder from the original volume of the cube.

The volume of the cylinder can be calculated using the formula V = πr^2h, where r is the radius and h is the height. In this case, r = 2cm and h = 14cm, so the volume of the cylinder is approximately 175.93cm^3.

Therefore, the new volume of the structure is 504cm^3 - 175.93cm^3 = 328.07cm^3.

Next, we need to calculate the new psi of the structure. Psi, or pounds per square inch, is a unit of pressure and can be calculated using the formula P = F/A, where P is the pressure, F is the force, and A is the area.

Since the only force acting on the structure is the air pressure, we can use the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

We know that the volume of the structure is 328.07cm^3 and the temperature is constant, so we can rearrange the ideal gas law to solve for pressure: P = nRT/V.

To determine the number of moles of gas, we can use the formula n = m/M, where n is the number of moles, m is the mass, and M is the molar mass. Since we are dealing with air, we can use the molar mass of air, which is approximately 28.97 g/mol.

The mass of the air inside the cube can be calculated using the density formula, d = m/V, where d is the density, m is the mass, and V is the volume. We know that the density of air is approximately 1.2 g/L, so the mass of the air inside the cube is (1.2 g/L)(0.504 L) = 0.6048 g.

Therefore, the number of moles of air inside the cube is (0.6048 g)/(28.97 g/mol) = 0.0209 mol.

Substituting this into the ideal gas law, we get P = (0