Consider the following depiction of a tank with radius r = 9 meters with a spout of height h = 4.5 meters.
A tank is half full of oil that has a density of 900 kg/m3. Find the work W required to pump the oil out of the spout. (Use 9.8 m/s2 for g and 3.14 for π. Round your answer to the nearest integer.)
To find the work required to pump the oil out of the spout, we can use the concept of work done against gravity.
The work W done against gravity is given by the formula: W = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height against which the object is lifted.
In this case, the height against which we are lifting the oil is the height of the spout, h = 4.5 meters.
To find the mass of the oil, we first need to calculate the volume of the oil in the tank. The tank is half full, so the volume of the oil in the tank is half the volume of the entire tank.
The volume of the tank can be calculated using the formula: V = πr^2h, where r is the radius of the tank and h is the height of the tank.
In this case, the radius of the tank is given as r = 9 meters, and the height of the tank is not provided. We will need to assume a height for the tank in order to calculate the volume. Let's assume a height of 10 meters for the tank.
The volume of the tank would then be V = π(9^2)(10) = π(81)(10) ≈ 2544 cubic meters.
Since the tank is half full, the volume of the oil is V/2 = 2544/2 = 1272 cubic meters.
To find the mass of the oil, we can multiply the volume by the density of the oil. The density of the oil is given as 900 kg/m^3.
The mass of the oil would then be m = density × volume = 900 × 1272 = 1,144,800 kg.
Now we can use this mass, along with the height of the spout, to calculate the work required to pump the oil out of the spout.
W = mgh = 1,144,800 × 9.8 × 4.5 = 49,989,360 N·m
Rounded to the nearest integer, the work required to pump the oil out of the spout is approximately 49,989,360 N·m.
what is the height of the tank? You need to know how far above the top of the oil is the spout, and how deep is the oil.
Anyway, assuming the oil has a depth of d meters, and the spout is h meters above the top of the oil, then consider the oil as a stack of thin slices of thickness dy.
then the volume of a slice of oil is 81π dy m^3. That means the weight of the oil is
900*81π dy N
The work is weight * distance, so the total work involved is
∫[0,d] 72900π (h-y) dy