What is the total force on the bottom of a 400-cm-diameter by 1.5-m-deep round wading pool due to the weight of the air and the weight of the water? (Note the pressure contribution from the atmosphere is 1.0 × 105 N/m2, the density of water is 1 g/cm3, and g = 9.8 m/s2.)
Water density = 1000 kg/m^3 which is 1 g/cm^3
Pa = 10^5 N/m^2 Pacals or N/m^2
additional pressure = rho g h = 1000 * 9.8 *1.5 = 9,800*1.5 = 14,700 Pascals
Total pressure down = 100,000 +14,700 = 114,700 Pascals
radius R = (400/100) /2 = 2 meters
pi R^2 = 3.14 * 4 = 12.56 m^3
total force down = 114,700 * 12.56 = 1,449,632 Newtons
(Remember that 100,000 Pascals of atmospheric also pushes up on the bottom from underneath. It is the 14,700 additional from the water that matters for design.)
To find the total force on the bottom of the wading pool, we need to consider two factors: the weight of the air and the weight of the water.
First, let's calculate the weight of the air. The pressure contribution from the atmosphere is given as 1.0 × 10^5 N/m^2. The weight of the air can be found by multiplying the pressure by the surface area of the bottom of the pool.
The surface area of the bottom of the pool can be calculated using the formula for the area of a circle: A = πr^2, where r is the radius of the pool. Given that the diameter is 400 cm, the radius is half of that, i.e., 200 cm or 2 meters.
So, the surface area of the bottom of the pool is A = π(2)^2 = 4π m^2.
Now, we can calculate the weight of the air by multiplying the pressure by the surface area: Weight of air = Pressure × Area = (1.0 × 10^5 N/m^2) × (4π m^2).
Next, let's calculate the weight of the water. The density of water is given as 1 g/cm^3. We need to convert this to kg/m^3, so we divide by 1000.
The height of the pool is given as 1.5 m. So, the volume of water in the pool can be calculated by multiplying the surface area of the bottom of the pool by the height: Volume of water = Area × Height = (4π m^2) × (1.5 m).
Now, we can calculate the weight of the water by multiplying the volume by the density and the acceleration due to gravity: Weight of water = Volume × Density × g = (4π m^2) × (1.5 m) × (1 kg/m^3) × (9.8 m/s^2).
Finally, to find the total force on the bottom of the wading pool, we sum the weight of the air and the weight of the water: Total force = Weight of air + Weight of water.
I hope you find this explanation helpful in calculating the total force on the bottom of the wading pool.