(a) Find the total force on the bottom of a cylindrical gasoline storage tank 15.0 m
high with radius 23.0 m. (b) Find the total force on the side of the tank.
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To find the total force on the bottom and side of the cylindrical gasoline storage tank, we need to use the principles of fluid pressure.
(a) Total force on the bottom of the tank:
The total force on the bottom of the tank is equal to the weight of the fluid above it. The weight of the fluid is determined by the density of the fluid and the height of the fluid column.
First, we need to find the weight of the fluid column above the bottom of the tank. The weight is given by the formula:
Weight = density * volume * g,
where density is the density of the fluid, volume is the volume of the fluid column, and g is the acceleration due to gravity.
Since the tank is cylindrical, the volume of the fluid column (V) can be calculated using the formula:
V = π * r^2 * h,
where r is the radius of the tank and h is the height of the fluid column.
Now we can calculate the weight of the fluid column. Let's assume the density of gasoline is 740 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2.
Weight = density * volume * g,
= 740 * π * (23)^2 * 15 * 9.8,
≈ 2,396,787.72 N.
Therefore, the total force on the bottom of the tank is approximately 2,396,787.72 Newtons.
(b) Total force on the side of the tank:
The total force on the side of the tank can be calculated by multiplying the pressure exerted by the fluid on the side wall by the surface area of the side wall.
The pressure exerted by the fluid at a certain depth (p) is given by the formula:
p = density * g * h,
where density is the density of the fluid, g is the acceleration due to gravity, and h is the vertical distance from the surface of the fluid to the point of interest.
Since the pressure acts perpendicular to the side wall, the surface area of the side wall of the tank is given by:
A = 2 * π * r * h,
where r is the radius of the tank and h is the height of the tank.
Now we can calculate the total force on the side of the tank. Assuming the height of the tank is 15.0 m and using the same density of gasoline (740 kg/m^3) and acceleration due to gravity (9.8 m/s^2):
p = density * g * h,
= 740 * 9.8 * 15,
= 108,810 Pa.
A = 2 * π * r * h,
= 2 * π * 23 * 15,
≈ 2167.5 m^2.
Total force on the side of the tank = p * A,
= 108,810 * 2167.5,
≈ 235,835,237.5 N.
Therefore, the total force on the side of the tank is approximately 235,835,237.5 Newtons.