A spherical hydrogen-filled balloon is used to lift a load of 5950 N (in addition to the weight of the hydrogen). Suppose the density of the air is 1.18 kg/m3. Find the minimum necessary radius for the balloon
To find the minimum necessary radius for the balloon, we need to consider the balance of forces acting on the balloon.
Let's denote:
R = radius of the balloon (unknown)
V = volume of the balloon
First, we need to determine the buoyant force acting on the balloon. The buoyant force is given by the difference in weight between the displaced air and the weight of the load.
The weight of the load is given as 5950 N, and we need to find the weight of the displaced air.
The weight of the hydrogen can be determined using its density and the volume of the balloon. The density of hydrogen is typically around 0.09 kg/m^3.
Weight of hydrogen = density of hydrogen * volume of balloon * gravitational acceleration
= (0.09 kg/m^3) * V * 9.8 m/s^2
The weight of the displaced air can be determined using the density of air and the volume of the balloon.
Weight of air = density of air * volume of balloon * gravitational acceleration
= (1.18 kg/m^3) * V * 9.8 m/s^2
Now, we can calculate the buoyant force:
Buoyant force = weight of displaced air - weight of hydrogen
= (1.18 kg/m^3) * V * 9.8 m/s^2 - (0.09 kg/m^3) * V * 9.8 m/s^2
= (1.18 - 0.09) * V * 9.8 m/s^2
The buoyant force must be equal to the weight of the load:
Buoyant force = 5950 N
Therefore, we have the equation:
(1.09) * V * 9.8 m/s^2 = 5950 N
Now, we can solve for V:
V = 5950 N / [(1.09) * 9.8 m/s^2]
Once we find V, we can calculate the radius of the balloon using the formula for the volume of a sphere:
V = (4/3) * π * R^3
Rearranging the formula, we get:
R = ∛[(3 * V) / (4 * π)]
Plug in the value of V to find the minimum necessary radius for the balloon.
To find the minimum necessary radius for the balloon, we can use Archimedes' principle, which states that the buoyant force acting on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.
1. Calculate the weight of the fluid displaced:
The weight of the fluid displaced can be calculated using the density of the fluid and the volume of the displaced fluid.
Given:
- Density of air: 1.18 kg/m^3
- Load weight: 5950 N
We need to convert the load weight from Newtons to kilograms:
Load weight = 5950 N
Convert N to kg: 1 N ≈ 0.101971621 kg
Load weight = 5950 N * 0.101971621 kg/N ≈ 605.285 kg
The weight of the fluid displaced is equal to the total weight (load weight + weight of hydrogen):
Weight of fluid displaced = Load weight + Weight of hydrogen
= 605.285 kg + Weight of hydrogen
2. Calculate the volume of the fluid displaced:
The volume of the fluid displaced can be calculated using the weight of the fluid displaced and the density of the fluid.
Given:
- Density of air: 1.18 kg/m^3
- Weight of fluid displaced: 605.285 kg
The volume of the fluid displaced can be calculated using the formula:
Volume of fluid displaced = Weight of fluid displaced / Density of air
Volume of fluid displaced = 605.285 kg / 1.18 kg/m^3 ≈ 513.822 m^3
3. Calculate the volume of the balloon:
The volume of the balloon is equal to the volume of the fluid displaced.
4. Calculate the radius of the balloon:
The volume of a sphere can be calculated using the formula:
Volume of a sphere = (4/3) * π * r^3
Rearranging the formula, we can solve for the radius:
(4/3) * π * r^3 = Volume of fluid displaced
r^3 = (3 * Volume of fluid displaced) / (4 * π)
r^3 ≈ (3 * 513.822 m^3) / (4 * π)
r^3 ≈ 388.090 m^3 / (4 * π)
r^3 ≈ 97.7215 / π
r ≈ ∛(97.7215 / π) ≈ 4.4518 m
Therefore, the minimum necessary radius for the balloon is approximately 4.4518 meters.