why does a balloon filled with helium float

Helium balloons are pulled by gravity, as are all objects with mass. The reason they don't fall is that there is another force acting on them, a buoyant force from air pressure that is equal to the weight of the air displaced by the balloon. The reason you don't float is that the weight of the air you displace is quite a bit less than your weight (a person is more dense than air). The reason a normal balloon doesn't float is that the weight of the air it displaces is just a little bit less than the weight of the balloon (because it is filled with air, but the rubber of the balloon itself is more dense than the air).


The analogy you want is to objects floating (or suspended) in water. Most rocks sink to the bottom, pulled by gravity, because the weight of the water they displace is less than their own weight. A bowling ball (ironically) is very close to the same density as water, so it will float suspended in mid-water, just like the helium balloon that has leaked a little bit.
On the left we have still air.


Imagine the volume of air inside the sphere has a weight W As the volume of air is not moving down due to gravity, its weight is balanced by the upwards force of the air surrounding it. U is this upwards force. (Upthrust) On the right, replace the same volume of air with helium. Helium is "lighter than air". That means less dense and less mass for the same volume. So its weight W is less than the weight of the same volume of air on the left.


However, the surrounding air hasn't changed and is still pushing up with a force U. This means that the upward force is now greater than the downwards force so the helium rises. In this example the weight of the skin of the balloon has been ignored. In practice, the upthrust would have to overcome this too in order to make the balloon rise. If you want to go deeper into the mechanism of how the upthrust comes about, then as others have said, the bottom of the balloon is at a greater depth in the atmosphere than the top.


This means the pressure of the air at the bottom is slightly greater than that at the top. As pressure is force per unit area, it means there is a greater force acting upwards (per unit area) on the bottom of the balloon than acting downwards on the top. Hence a net upwards force. This upwards force is opposed by the weight of the balloon downwards. It's the relative sizes of these two forces which determines whether the balloon rises, falls or stays still.

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