Find the force on a hollow dome that is 1000 feet in diameter, that is a mile below the surface. HOw large can the dome be expanded and not surpass 600 billion pounds.
I know the height will be 500, and i think that the intergral is 5280 to 4780 int 64,000pih, I am not sure thoug. any help would be much appreciated.
Is this in water on earth?
Do you mean the buoyant force (the net force of water on the sphere)?
If so, use Archimedes principle. The force up is equal to the weight of water displaced.
= rho g (4/3) pi r^3
To find the force on a hollow dome, we need to calculate the weight of the water above the dome. The weight of the water can be determined by integrating the pressure over the surface area of the dome.
1. First, let's calculate the pressure at a depth of a mile (5280 feet) below the surface. The pressure at a given depth is given by P = ρgh, where P is the pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the depth.
The density of water, ρ, is approximately 62.4 pounds per cubic foot.
The acceleration due to gravity, g, is approximately 32.2 feet per second squared.
Plugging in these values, we get P = 62.4 * 32.2 * 5280 = 10,621,184 pounds per square foot.
2. Next, we need to find the surface area of the dome. The dome is a hollow shape, so we need to subtract the area of the inner circle (diameter of 1000 feet) from the area of the outer circle (diameter of 1500 feet) to get the surface area of the dome.
The area of a circle is given by A = πr^2, where A is the area and r is the radius.
The radius of the inner circle is 500 feet, and the radius of the outer circle is 750 feet.
The surface area of the dome is then A = π(750^2 - 500^2) = 250π(750 + 500) = 750,000π square feet.
3. Now, we can calculate the force on the dome by multiplying the pressure by the surface area: Force = Pressure * Surface Area.
Force = 10,621,184 pounds per square foot * 750,000π square feet = 7,965,888,000,000π pounds ≈ 25 trillion pounds.
From the calculations, the force on the dome is approximately 25 trillion pounds. To keep the force below 600 billion pounds, we need to find out how much larger the dome can be expanded.
Let's denote the maximum radius the dome can be expanded by as r_max. We can find r_max by rearranging the formula for the force on the dome:
Force = Pressure * Surface Area
600 billion pounds = 10,621,184 pounds per square foot * π * r_max^2
Solving for r_max:
r_max^2 = (600 billion pounds) / (10,621,184 pounds per square foot * π)
r_max ≈ √(56.52926063) = 7.523 feet
Therefore, the dome can be expanded by approximately 7.523 feet in radius (or 15.046 feet in diameter) without surpassing the limit of 600 billion pounds.