Question

On October 21, 2001, Ian Ash pole of the United Kingdom achieved a record altitude of $\mathbf{\text{3.35km}}\mathbf{\left(}\mathbf{\text{11000ft}}\mathbf{\right)}$powered by $\mathbf{600}$toy balloons filled with helium. Each filled balloon had a radius of about $\mathbf{0}\mathbf{.50}\mathbf{\text{m}}$and an estimated mass of $\mathbf{\text{0.30kg}}$ . (a) Estimate the total buoyant force on the $\mathbf{600}$balloons. (b) Estimate the net upward force on all $\mathbf{600}$balloons. (c) Ash pole parachuted to the Earth after the balloons began to burst at the high altitude and the buoyant force decreased. Why did the balloons burst?

Short answer

(a) The total buoyant force on the $600$balloons is $\text{1.9kN}$.

(b) The net upward force on all $600$balloons is $\text{1.9kN}$.

(c) Atmospheric pressure at this high altitude is much lower than at Earth’s surface, so the balloons expanded and eventually burst.

Step-by-step solution

A concept:

When an object is partially or fully submerged in a fluid, the fluid exerts on the object an upward force called the buoyant force.

According to Archimedes’s principle, the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object:

$\mathit{B}\mathbf{=}{\mathit{\rho }}_{\mathbf{a}\mathbf{i}\mathbf{r}}{\mathit{V}}_{\mathbf{o}\mathbf{b}\mathbf{j}\mathbf{e}\mathbf{c}\mathbf{t}}\mathit{g}$

Where, $\mathit{B}$is the buoyant force,${\mathit{\rho }}_{\mathbf{air}}$is the density of the air,$\mathit{g}$gravitational acceleration, and${\mathit{V}}_{\mathbf{object}}$volume displaced.

(a) Estimate The total buoyant force on the 600 balloons:

The total buoyant force of the $600$toy balloons is define as below.

$\begin{array}{rcl}{B}_{total}& =& 600\times B_{\sin glebaloon}\\ & =& 600\times \left({\rho }_{air}g{V}_{baloon}\right)\\ & =& 600\times \left[{\rho }_{air}g\times \frac{4}{3}\pi r^3\right]\end{array}$

$\begin{array}{rcl}{B}_{total}& =& 600\left[\left(1.20\times {10}^3\raisebox{1ex}{$\text{kg}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^{\text{3}}$}\right.\right)\left(9.8\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{${\text{s}}^{\text{2}}$}\right.\right)\times \frac{4}{3}\times 3.14\times {\left(1.5\text{0m}\right)}^3\right]\\ & =& 3.7\times 1{\text{0}}^{\text{3}}\text{N}\\ & =& 1.\text{9kN}\end{array}$

(b) The net upward force on all 600 balloons:

Estimate the net upward force by applying Newton’s second law in the vertical direction:

$\begin{array}{rcl}\sum F_y& =& B_{total}-m_{total}g\\ & =& 3.7\times {10}^3\text{N}-600\left(0.\text{30kg}\right)\left(\text{9.8}\raisebox{1ex}{${\displaystyle \text{m}}$}\!\left/ \!\raisebox{-1ex}{${\text{s}}^{\text{2}}$}\right.\right)\\ & =& 1.9\times {10}^3\text{N}\\ & =& 1.\text{9kN}\end{array}$

This net force was sufficient to lift Ash pole, his parachute, and other supplies.

(c) Atmospheric pressure at this high altitude:

Atmospheric pressure at this high altitude is much lower than at Earth’s surface, so the balloons expanded and eventually burst.