A cylindrical slime tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is 6 feet and the slime weighs 54 pounds per cubic feet?
an excellent presentation on this topic, with a video, even, is at
http://www.sophia.org/tutorials/fluid-force-integral-calculation
Plug in your numbers, and c'mon back if you get stuck. And show how far you got.
I did the problem.
I found the fluid force to be 54 times the integral from -3 to 0 of (-y)(2)(root(9-y^2)dy
my final answer was 972 pounds. Is that correct?
note from my work:
h(y)=-y
L(y)=2x=2root(9-y^2)
To find the fluid force on a circular end of the tank, we need to calculate the pressure exerted by the fluid and then multiply it by the area of the circular end.
1. Calculate the radius of the cylinder:
The diameter is given as 6 feet, so the radius is half of that: 6 feet / 2 = 3 feet.
2. Calculate the height of the fluid:
The tank is half full, so the height of the fluid is half of the height of the cylinder. Since no height is given, we assume the height of the cylinder is the same as its diameter (6 feet).
Therefore, the height of the fluid is 6 feet / 2 = 3 feet.
3. Calculate the pressure exerted by the fluid:
The pressure is given by the formula P = ρgh, where:
- P is the pressure (in pounds per square foot)
- ρ is the density of the fluid (in pounds per cubic foot)
- g is the acceleration due to gravity (approximately 32.2 feet per second squared)
- h is the height of the fluid (in feet)
Plugging in the values, we get:
P = (54 lb/ft^3) x (32.2 ft/s^2) x (3 ft)
P = 5210.4 lb/ft^2
4. Calculate the area of the circular end:
The area of a circle is given by the formula A = πr^2, where:
- A is the area (in square feet)
- π is a constant (approximately 3.14159)
- r is the radius (in feet)
Plugging in the values, we get:
A = π x (3 ft)^2
A = π x 9 ft^2
A ≈ 28.27 ft^2
5. Calculate the fluid force on the circular end:
The fluid force is given by the formula F = PA, where:
- F is the fluid force (in pounds)
- P is the pressure (in pounds per square foot)
- A is the area (in square feet)
Plugging in the values, we get:
F = (5210.4 lb/ft^2) x (28.27 ft^2)
F ≈ 147,223 lb
Therefore, the fluid force on a circular end of the tank is approximately 147,223 pounds.
To find the fluid force on a circular end of the tank, we need to first calculate the volume of the slime in the tank, and then multiply it by the weight density of the slime to find the force.
Given that the diameter of the tank is 6 feet, we can find the radius by dividing the diameter by 2: r = 6 ft / 2 = 3 ft.
Since the tank is half full, the height of the slime will be equal to the radius: h = 3 ft.
The volume of the slime can be calculated using the formula for the volume of a cylinder: V = π * r^2 * h. Plugging in the values, we get: V = π * (3 ft)^2 * 3 ft = 27π cubic ft.
Next, we multiply the volume of the slime by the weight density of the slime to find the fluid force: F = V * density.
Given that the slime weighs 54 pounds per cubic foot, we have: F = 27π cubic ft * 54 lb/ft^3 = 1458π lb.
Therefore, the fluid force on the circular end of the tank is 1458π pounds.