An oil tank in the shape of a right circular cylinder, with height 30 meters and a radius of 5 meters is two-thirds full of oil. How much work is required to pump all of the oil over the top of the tank? (The density of oil is 820 kg/m^3 ).
To find the work required to pump all of the oil over the top of the tank, we need to calculate the volume of the oil.
First, let's find the volume of the whole tank.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
Plugging in the values, we get V = π(5^2)(30) = 750π m^3.
Since the tank is two-thirds full, we need to find the volume of two-thirds of the tank.
Volume of two-thirds of the tank = (2/3)(750π) = 500π m^3.
Now, let's find the mass of the oil.
The mass of an object is given by the formula Mass = density × volume.
Plugging in the value of the density (820 kg/m^3) and volume (500π m^3), we get Mass = 820 × 500π kg.
Finally, let's calculate the work required to pump all of the oil.
The work done is given by the formula W = force × distance.
Since the force required to lift an object is equal to the weight of the object, we need to multiply the mass of the oil by the acceleration due to gravity.
The acceleration due to gravity is approximately 9.8 m/s^2.
So, W = (820 × 500π) × 9.8 joules.
The work required to pump all of the oil over the top of the tank is approximately 404,736π joules.
To calculate the work required to pump all the oil over the top of the tank, we need to determine the volume of the oil and then use the formula for work.
First, let's find the volume of the oil in the tank. The tank is two-thirds full, which means the oil fills 2/3 of the total volume of the cylinder.
The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height.
In this case, the radius (r) is 5 meters and the height (h) is 30 meters. Therefore, the volume of the entire cylinder is:
V_cylinder = π * (5^2) * 30
V_cylinder = 2250π cubic meters
Since the tank is two-thirds full, the volume of the oil is:
V_oil = (2/3) * V_cylinder
V_oil = (2/3) * 2250π
V_oil = 1500π cubic meters
Now, let's find the mass of the oil. The density of oil is given as 820 kg/m^3.
The formula for mass is m = density * volume. Therefore, the mass of the oil is:
m = 820 * V_oil
m = 820 * 1500π
m ≈ 1,230,000π kg
Finally, we can calculate the work required to pump all the oil over the top of the tank. The formula for work is W = force * distance.
In this case, force is equal to the weight of the oil and distance is equal to the height of the tank, since we are pumping the oil to the top.
The weight of the oil is given by the formula weight = mass * gravity. Here, gravity is 9.8 m/s^2.
Therefore, the work required to pump the oil is:
W = weight * distance
W = (m * g) * h
W = (1,230,000π * 9.8) * 30
W ≈ 3,593,400π Joules
So, approximately 3,593,400π Joules of work is required to pump all the oil over the top of the tank.
The weight of oil in the tank is
w = (20*25π m^3) * (820 kg/m^3) * 9.8 = 4018000πN
The center of gravity of the oil is 20m from the top of the tank, so the work required to empty the oil is
4018000π * 20 = 80360000π J
Or, using calculus, the weight of oil in a thin layer of thickness dy is
25π dx * 820 * 9.8 = 200900π N
To lift all the oil, then the work is
∫[10,30] 200900π y dy = 80360000π J