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 x 10^5 N/m^2, the density of water is 1 g/cm^3, g=9.8m/s^2.)
(Answer: 1.44 x 10^6 N)
To calculate the total force on the bottom of the wading pool, we need to consider the weight of the air and the weight of the water pushing down on it.
First, let's determine the weight of the air. The pressure contribution from the atmosphere is given as 1.0 x 10^5 N/m^2. Since pressure is defined as force per unit area, we can treat this value as the force per unit area exerted by the air on the pool's bottom.
The area of the pool's bottom can be calculated using the diameter. The diameter is given as 400 cm, which means the radius is half of that (200 cm or 2 m). The area of a circle can be calculated using the formula A = πr^2. Hence, the area is:
A = π x (2 m)^2
A = 4π m^2
Now, we can calculate the force exerted by the air on the bottom of the pool:
F_air = P x A
F_air = (1.0 x 10^5 N/m^2) x (4π m^2)
Next, let's determine the weight of the water. The density of water is given as 1 g/cm^3, which can be converted to 1000 kg/m^3 (since 1 g = 0.001 kg).
The volume of the pool can be calculated using the diameter and the depth. The volume of a cylinder can be calculated using the formula V = πr^2h. Hence, the volume is:
V = π x (2 m)^2 x (1.5 m)
V = 6π m^3
Now, we can calculate the weight of the water as follows:
W_water = density x volume x g
W_water = (1000 kg/m^3) x (6π m^3) x (9.8 m/s^2)
Finally, to find the total force on the bottom of the pool, we need to sum the forces:
Total force = F_air + W_water
Now, you can plug in the values into the formulas and calculate the answer:
Total force = (1.0 x 10^5 N/m^2) x (4π m^2) + (1000 kg/m^3) x (6π m^3) x (9.8 m/s^2)
After evaluating the expression, you should obtain the answer of 1.44 x 10^6 N.
To find the total force on the bottom of the wading pool, we need to consider the weight of the air and the weight of the water.
1. Calculate the weight of the water:
The density of water is given as 1 g/cm^3, which is equivalent to 1000 kg/m^3 when converted to SI units.
The volume of the wading pool can be calculated using the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.
Given that the pool has a diameter of 400 cm, the radius is half of that: r = 200 cm = 2 m.
The height of the pool is given as 1.5 m.
Using the formula, we find: V = π(2^2)(1.5) = 6π m^3.
The weight of the water is then calculated by multiplying the volume by the density and the acceleration due to gravity:
Weight of water = Volume × Density × g = 6π × 1000 × 9.8 N.
2. Calculate the weight of the air:
The weight of air is calculated using the formula: Weight of air = Pressure × Area, where the pressure is given as 1.0 x 10^5 N/m^2 and the area is the same as the bottom area of the pool.
The radius of the pool is 2 m, so the area is given by: Area = πr^2 = π(2^2) m^2.
The weight of the air is then calculated by multiplying the pressure by the area:
Weight of air = 1.0 x 10^5 × π(2^2) N.
3. Find the total force on the bottom of the wading pool by summing the weights of the water and the air:
Total force = Weight of water + Weight of air.
Plugging in the values and calculating:
Total force = (6π × 1000 × 9.8) + (1.0 x 10^5 × π(2^2))
Total force = 58,800π + 40,000π
Total force = 98,800π
Using the approximation π ≈ 3.14:
Total force ≈ 98,800 × 3.14
Total force ≈ 310,352 N
Therefore, the total force on the bottom of the wading pool due to the weight of the air and the weight of the water is approximately 310,352 N.
using Newtons, kg, meters
water pressure = 1000 kg/m^3*9.8 m/s^2 * 1.5 m = 14,700 N/m^2
so total pressure = 14,700 +100,000 = 114,700 N/m^2
area = pi d^2/4 = pi 4^2/4 = 12.6 m^2
s total force down = 114,700 *12.6
= 1441400 = 1.44*10^6 sure enough
Note, the air also pushes up on the bottom of the pool so things are not as bad as they look