If you live in a city your stove can be powered by natural gas. This gas is mainly methane gas and has a density of about 0.72 kilograms per cubic meter. What minimum sized balloon would you have to fill in order to be able to be lifted into the air? Take your mass to be 75 kilograms and the temperature of the outside air to be 20 degrees C. Neglect the mass of the balloon rubber and assume that the balloon is spherical in shape.
To determine the minimum sized balloon required to lift your mass into the air, we need to consider the buoyant force acting on the balloon. The buoyant force is equal to the weight of the air displaced by the balloon.
To calculate the buoyant force, we can use the formula:
Buoyant force = (Density of air) * (Volume of balloon) * (Acceleration due to gravity)
First, let's find the volume of the balloon:
Mass of air = Density of air * Volume of balloon
Since the balloon is spherical in shape, we can use the formula for the volume of a sphere:
Volume of sphere = (4/3) * π * (radius of sphere)^3
We don't know the radius of the sphere yet, so we'll need to calculate that.
The mass of air in the balloon would be equal to the mass of methane gas required to achieve neutral buoyancy. This can be calculated using the mass of the atmosphere displaced given by the equation:
Mass of air displaced = mass of the balloon
Now we can rewrite the equation for the mass of the air in the balloon as:
Mass of air = (Density of air in balloon) * (volume of balloon)
Setting the equations for mass of air equal to each other:
(Density of air in balloon) * (Volume of balloon) = (Density of air) * (Volume of balloon)
Density of air in balloon = Density of air
Now, back to finding the radius of the balloon:
Mass of air = Density of air * Volume of balloon
(0.72 kg/m^3) * (4/3) * π * (radius of sphere)^3 = (75 kg)
Simplifying, we find:
(4/3) * π * (radius of sphere)^3 = (75 kg) / (0.72 kg/m^3)
(4/3) * π * (radius of sphere)^3 ≈ 130.2 m^3
To find the radius of the sphere, we can solve for the radius:
(radius of sphere)^3 ≈ (130.2 m^3) / ((4/3) * π)
(radius of sphere)^3 ≈ 98.2
Taking the cube root of both sides, we find:
radius of sphere ≈ 4.4 m
Thus, the minimum sized balloon required to lift your mass into the air would have a radius of approximately 4.4 meters.
To determine the minimum size of the balloon needed to lift your weight, we can use Archimedes' principle, which states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.
To calculate the buoyant force, we need to know the density of the surrounding air and the volume of the balloon.
1. Convert the density of methane gas from kilograms per cubic meter to grams per liter:
0.72 kg/m^3 = 720 g/m^3.
2. Calculate the density of air:
The density of air at 20 degrees Celsius is approximately 1.204 kg/m^3 or 1.204 g/L.
3. Calculate the apparent weight of the system:
Apparent weight = Mass x gravity.
The mass is given as 75 kg, and gravity is approximately 9.8 m/s^2.
Apparent weight = 75 kg x 9.8 m/s^2 = 735 N.
4. Calculate the volume of the displaced air:
Volume of displaced air = Apparent weight / Density of air.
Volume of displaced air = 735 N / (1.204 kg/m^3 x 9.8 m/s^2) ≈ 61.5 m^3.
5. Calculate the radius of the balloon:
Assuming a spherical shape, the volume of a sphere is given by:
Volume = (4/3) x π x r^3.
Rearranging the equation to solve for the radius:
r = (3 x Volume / (4 x π)) ^(1/3).
r = (3 x 61.5 m^3 / (4 x 3.14159)) ^(1/3) ≈ 3.06 meters.
6. Calculate the diameter of the balloon:
Diameter = 2 x radius = 2 x 3.06 meters ≈ 6.12 meters.
Therefore, the minimum-sized balloon you would need to fill in order to lift your weight would have a diameter of approximately 6.12 meters.