Suppose that a party balloon is spherical, has a radius of 16 centimetres and is filled with helium. Furthermore suppose that we do our experiment under standard atmospheric conditions, when the air density is ρ=1.225kg/m3 . How many of these balloons should a 75 kilogram human carry in order to become airborne?
you want the volume to displace 75 kg of air.
So, that would be 61.22 m^3 of air
Using PV=kT, with P=1, pick a reasonable temperature and find V
To determine the number of balloons required to lift a 75-kilogram human, we need to consider the buoyant force exerted by the balloons on the person. The buoyant force is equal to the weight of the air displaced by the balloons, which is given by the formula:
Buoyant Force = Volume of Displaced Air x Air Density x gravitational acceleration
Let's break down the calculation step by step:
1. Calculating the volume of the balloon:
The volume of a sphere is given by the formula: V = (4/3) x π x r^3, where r is the radius of the sphere. In this case, the radius of the balloon is 16 centimeters (0.16 meters).
V = (4/3) x π x 0.16^3
V = 0.03409 m^3 (approximately)
2. Calculating the weight of the displaced air:
The weight of the displaced air is equal to the volume of air displaced multiplied by the air density (ρ).
Weight of Displaced Air = Volume of Displaced Air x Air Density
Weight of Displaced Air = 0.03409 m^3 x 1.225 kg/m^3
Weight of Displaced Air = 0.04177 kg (approximately)
3. Calculating the buoyant force:
The buoyant force is equal to the weight of the displaced air multiplied by the gravitational acceleration (g).
Buoyant Force = Weight of Displaced Air x gravitational acceleration
Buoyant Force = 0.04177 kg x 9.8 m/s^2
Buoyant Force = 0.40919 N (approximately)
4. Calculating the number of balloons needed:
To become airborne, the buoyant force exerted by the balloons should be greater than or equal to the weight of the person (75 kg) multiplied by the gravitational acceleration.
Number of Balloons = (Weight of Person x gravitational acceleration) / Buoyant Force
Number of Balloons = (75 kg x 9.8 m/s^2) / 0.40919 N
Number of Balloons = 179.98 (approximately)
Therefore, a person weighing 75 kilograms would need approximately 180 of these balloons to become airborne. Since we can't have a fraction of a balloon, the person would need to round up to the nearest whole number, so the answer is 180 balloons.