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 100 kilograms and the temperature of the outside air to be 20 degrees C. Neglect the mass of the balloon rubber and basket and assume that the balloon is spherical in shape.Determine the minimum diameter of this balloon. Hint: review your notes on buoyant forces.
You do not say what is in the balloon. Is it Helium, Hydrogen, or hot air?
In any case use Archimedes who is very good at lifting things.
Force up = volume of air displaced times density of air times gravity, (4/3)pi r^3 * density of air at 20 degrees C * g
Weight of interior gas down = same equation but use density of interior gas
so in the end
100 * g = (4/3) pi r^3 [ density air - density gas] * g
Notes :
g cancels
that gives you r, multiply by 2 for d
the gas is mainly methane gas and has a density of about 0.72 kilograms per cubic meter so you know that a balloon filled with natural gas will float because the density is lower than air
Those are the options.I don't get any of those in my calculation but ill continue to try it. Thanks
about 8.5 meters in diameter
about 10.75 meters in diameter
about 7.3 meters in diameter
about 9.2 meters in diameter
about 7.6 meters in diameter
To determine the minimum diameter of the balloon needed to lift a 100-kilogram person into the air, we can use the concept of buoyant force.
The buoyant force is the force exerted on an object immersed in a fluid, in this case, air. It is equal to the weight of the fluid displaced by the object. In order to lift a person, the buoyant force must be greater than or equal to the person's weight.
Let's break down the problem:
1. Determine the weight of the person:
The weight of the person is given as 100 kilograms, but we need to convert it to Newtons (weight unit in SI system) using the formula W = mg, where g is the acceleration due to gravity (approximately 9.8 m/s²):
Weight = 100 kg * 9.8 m/s² = 980 N
2. Calculate the volume of air displaced by the balloon:
Since the balloon is assumed to be spherical, we can use the formula for the volume of a sphere:
V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
Since the diameter (d) of the sphere is what we need, we can rewrite the formula as:
V = (4/3)π(d/2)³ = (1/6)πd³
3. Calculate the weight of the displaced air:
Since air has a density of approximately 1.2 kg/m³, we can use the formula W = mg to find the weight of the displaced air:
Weight of air = Density of air * Volume of air * g
Weight of air = 1.2 kg/m³ * V * 9.8 m/s²
Weight of air = 11.76 V
4. Set up the equilibrium condition:
The buoyant force must be equal to or greater than the weight of the person, so we have:
Buoyant force = Weight of air ≥ Weight of the person
11.76 V ≥ 980 N
5. Solve for the minimum diameter:
Using the volume formula V = (1/6)πd³, we can rearrange the previous equation to solve for d:
(1/6)πd³ ≥ 980 / 11.76
(1/6)πd³ ≥ 83.33
d³ ≥ 83.33 * (6/π)
d³ ≥ 83.667
d ≥ ∛(83.667)
Calculating the cube root gives us a minimum diameter of approximately 4.675 meters.
So, the minimum diameter of the balloon needed to lift a 100-kilogram person in the given conditions is approximately 4.675 meters.