pressure - The physics behind Balloons lifting objects?

Apologies for the super basic question, but we all have to start somewhere right?

Can somebody please explain exactly how you would calculate the number of helium balloons it would take to lift an object of mass $m$ here on earth, the variables I would need to take into account and any other physics that come into play.

I think I can roughly calculate it using the method below but would love somebody to explain how this is right/wrong or anything I have negated to include. This model is so simple, I am thinking it can't possibly be correct.

• 1 litre of helium lifts roughly 0.001kg (I think?)


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  Assumption: an inflated balloon is uniform and has a radius $r$ of 0.1m

  
  


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  $\frac{4}{3}\pi r^3 = 4.189$ cubic metres $\approx$ 4 litres capacity per balloon

  
  



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  Lets say $m = 1$kg, therefore $\frac{m\div0.001}{4} = 250$ balloons to lift that object?

  
  

As you can tell, I haven't touched Physics since high school and would really appreciate any help. It seems like an easy question, but actually is probably more complex than I thought.

Thanks a lot.

Answer

The net upward force is, according to Wiki buoancy:

$$F_\mathrm{net}=\rho_\mathrm{air}V_\mathrm{disp}g-m_\mathrm{balloon} \cdot g$$ For helium, the $m_\mathrm{balloon}=\rho_\mathrm{helium}V_\mathrm{disp} + m_{shell}$, thus $$F_\mathrm{net}=\rho_\mathrm{air}V_\mathrm{disp}g-\left(\rho_\mathrm{helium}V_\mathrm{disp} + m_{shell} \right)\cdot g=\left(\rho_{air}-\rho_\mathrm{helium}\right)V_\mathrm{disp} \cdot g - m_{shell} \cdot g$$ With $V_\mathrm{disp}=N_\mathrm{balloon} V_\mathrm{balloon}$ and $F_\mathrm{net}=m_\mathrm{load} \cdot g$, you're able to calculate the number of balloons necessary.

EDIT: Some more steps how to actually solve the problem.

To isolate the value of $N_\mathrm{balloon}$, we plug in the volume expression to obtain:

$$F_\mathrm{net}=\left(\rho_{air}-\rho_\mathrm{helium}\right)N_\mathrm{balloon} V_\mathrm{balloon} \cdot g - m_\mathrm{shell} \cdot g$$

We can then isolate the value of $N_\mathrm{balloon}$ by adding $m_\mathrm{shell} \cdot g$ on both sides:

$$F_\mathrm{net} + m_\mathrm{shell} \cdot g=\left(\rho_{air}-\rho_\mathrm{helium}\right)N_\mathrm{balloon} V_\mathrm{balloon} \cdot g$$

And then divide both sides by $\left(\rho_{air}-\rho_\mathrm{helium}\right)V_\mathrm{balloon} \cdot g$ to get:

$$\frac{F_\mathrm{net} + m_\mathrm{shell}\cdot g}{\left(\rho_{air}-\rho_\mathrm{helium}\right)V_\mathrm{balloon} \cdot g}=N_\mathrm{balloon}$$