Pressure, buoyancy, and gravity applied to a balloon





  • Assume the pressure inside and outside the balloon is equal at 1 atmosphere.
  • For ease of calculation, the volume and mass of the balloon and string are assumed to be zero.

Pressure on a Balloon

In the above simulation, the pressure inside and outside the balloon was assumed to be the same at 1 atmosphere.
However, in reality, the internal pressure is slightly higher due to the elasticity of the rubber forming the balloon.
In other words, Internal pressure = External pressure + Pressure due to membrane tension.

The Young–Laplace equation describes the pressure inside and outside the balloon. If the pressure inside the balloon is \(P_{in}\) and the pressure outside is \(P_{out}\), it can be approximated by the following equation:
\[P_{in} = P_{out} + \frac{2T}{r}\]
In the equation above, \(T\) is the surface tension of the balloon membrane, and \(r\) is the radius of the balloon. This equation has the following meaning:

  1. A balloon with less gas has a thick rubber membrane, resulting in high surface tension and a small radius. Consequently, the internal pressure exceeds the external pressure. → In reality, balloons are difficult to inflate initially.
  2. Therefore, as the balloon contains a certain amount of gas, the difference between the internal and external pressures decreases, making it relatively easy to inflate.
  3. As the balloon grows larger and reaches its elastic limit, surface tension increases, causing the internal pressure to increase again.

Force applied to the balloon

Float balloon

All objects located on the Earth’s surface are acted upon by gravity, which pulls them toward the center of the Earth (downward). On the other hand, they also receive an upward buoyant force equal to the amount of air displaced.
By calculating each of these forces and combining them, we can determine the net force acting on the balloon.

1. Buoyancy
In the above simulation, buoyancy is the weight of the air displaced by the balloon.

Buoyancy(N) = 9.8 × volume of balloon(L) × density of the atmosphere(kg/L)

2. Gravity
The force of gravity can be obtained by multiplying the mass of the balloon by 9.8.

Gravity(N) = 9.8 × volume of balloon(L) × density of balloon(kg/L)

Conclusion

When the buoyant force exceeds gravity, the balloon experiences an upward force.
Conversely, when gravity is greater than the buoyant force, the balloon experiences a downward force.