Every child has had that heartbreak of seeing their helium balloon escape their grasp and drift off into the stratosphere. Many of us dreamt of flying up with those balloons and hoped that if we held enough maybe we’d start to feel something. But how many balloons exactly would I need to fly?
You may have heard of Archimedes as the crazy greek guy who got in the bath, saw the water went up and then ran down the street screaming “EUREKA”, but in fact, he was a very clever mathematician and physicist whose theories are still taught today. One of those laws is going to be very important in this example, the force of buoyancy on a body is equal to the weight of the fluid it displaces. This is important because air is a fluid, therefore, anything in the air will have a force of buoyancy.
Newton’s Second Law
Newton’s laws are probably the most important discovery in physics. They tell us how everything moves and how forces work. What we need for this test is Newton’s second law which tells us that the resultant force is equal to the sum of all the other forces. What that means is that if we have one force to the left of 5N and one to the right of 6N, are resultant force will be 1N to the right as the other 5N cancel out. Why this is important for this is, it tells us that in order for us to accelerate upwards and ‘fly’, we need a bigger force upwards than we do downwards. In this case, our force upwards would be our buoyancy force from Archimedes Principle and our downwards force is my weight.
Normally, if we want balloons to float we fill them with helium, but why? Well, let’s use Archimedes principle. To start off with, we need the volume of the balloon, if we say it is a sphere we can find the volume using the formula V=(4/3)*Π*r^3, where r is the radius, given a radius of 15cm, that gives us about 0.014 cubic meters. At sea level and at 15°C, air has a density of 1.225kg per cubic meter, to find the weight of air displaced we need to multiply 1.225 by 0.014 by 9.8, which gives us a buoyancy force lifting us up of 0.168N. But we can’t stop there because a balloon has mass, and if it has mass and it’s on earth, it has weight. Let’s say the mass of the latex of the balloon is 0.001kg, now the mass of the helium in the balloon is equal to its density (0.164 kg per cubic meter) multiplied by the volume occupied (0.014 cubic meters), this gives us a mass of 0.0023kg. This means that overall the balloon has a mass of 0.0033kg, giving it a weight of 0.0323N. So, using Newton’s second law, we have a resultant force upwards of 0.168 – 0.0324 = 0.1357N.
Lift me up
So now we know that each balloon is giving an upwards force of 0.1357N, but how many do we need to lift me up. To start with I am also displacing air, that means there is some buoyancy force on me. The average human has a volume of 70 cubic meters, this means that I have a buoyancy force of 0.84N. I have a mass of 80kg, this means I have a weight of 784N, so far this means that we have 1 downward force of 784N, one upwards of 0.84N, giving a resultant of 783.16N downwards. I will start to float once the resultant force is upwards, this means we need enough balloons to make an upwards force of 783.16N. With each balloon giving an upwards force of 0.1357N, that means we need 5772 balloons to make an upwards force of 783.16N. This means that we need 5772 standard party balloons for me to start, albeit extremely slowly, floating upwards. If you wanted to try this at home, to get enough helium you would need to pay around £2000 and with the world coming to a helium shortage, I wouldn’t recommend it.