Finding the number of ballons needed to make a child fly away

[arhzz]

Homework Statement

A child m = 35 kg wants to buy balloons so that he can take off. How many balloons with a radius of 0.25 m full of helium,does he need to buy so that he can float away.The thickness of helium is 0,178 kg/m^3 and the thickness of the air is 1,29 kg/m^3 The ballons are shaped as a sphere

Relevant Equations

Lift equation

Hello! So my thought process is in order to achive lift the lift force has to equal the force that is pulling the child to the ground.So the force pulling the child down we can calculate like this

$$ F = mg $$ S
F should be 343,35 N

Now to calculate the lift I've used this formula $$ p = \frac{F}{A} $$ Now we can write F like this $F = \rho * g * V$

Now here is where I am kind of having a dilema.I'm prett sure I need to calculate the Volume and Area of the balon (probably why the r was given) and I've done that;

$$ V = 0,065 m^3$$
$$ A = 0,785 m^2$$

Now if i put these back into my equation for p I get p = 0,144 N (assuming I've done everything correctly in the calculation). Now obviously this isn't really what I was expecting to happen.Where its the flaw in my logic? I've also noticed I haven't used the thickness of air.By logic it should also play a role into the lift force but I'm not really able to wrap my head around all of this. Any help would be great.Thank you!

 

[Orodruin]

I'm prett sure I need to calculate the Volume and Area of the balon

Why?
What are you going to use the area/volume for? What is your idea for solving for the number of balloons?

Note that $F = pA$ is only valid on a planar surface with a force direction that is normal to the surface. If you have a curved surface you need to take into account that the direction of the force varies depending on where on the surface you are as forces add like vectors.

Edit: Are you familiar with Archimedes' principle?

 

[haruspex]

Ask yourself, why does a Helium balloon rise at all?
(Btw, the thread question is an important topic of research. Many adults would like to know the answer.)

 

[phinds]

(Btw, the thread question is an important topic of research. Many adults would like to know the answer.)

Especially a month or two into a quarantine.

Rambunctious little boy: Mommy, am I adopted?
Mom: Not yet sweety, I just put the ad in Craig's List yesterday.

 

[arhzz]

I am familiar with the archimdes principle (not too deeply) but I though it only applied to water? Now I thought that I needed the area/volume to calculate the p (lift). And how would I solve for the number of ballons is a good question.I think if I get the problem going an idea would come up but right now I'm not sure how.

 

[Orodruin]

I am familiar with the archimdes principle (not too deeply) but I though it only applied to water?

Archimedes' principle applies to any object immersed in a fluid.

Now I thought that I needed the area/volume to calculate the p (lift).

Hence, the question about why you thought this. What was your thought process? Did you have a reason to think so or were you just guessing? We have discussed Archimedes' principle - how would you apply it?

 

[arhzz]

Archimedes' principle applies to any object immersed in a fluid.Hence, the question about why you thought this. What was your thought process? Did you have a reason to think so or were you just guessing? We have discussed Archimedes' principle - how would you apply it?

Oh okay,now Archimdes makes more sence.

Yea I though if I was going to calculate the lift,the formula requiers area and volume.Simple as that.I'll check on archimedes here need to read through some literature.

 

[arhzz]

Okay so I've taken my sweet time read through some literature my notes and this is what I've got so far.

So if we want to find what the force of lift is,we should note that it is equall to the weight of the compressed fluid so.

$$ F_{lift} = m_{air} * g $$

The mass of air can be calculated as $m = \rho * V$ where V is the volume of the sphere shaped baloon.(the 0,065m^3 I've calculated in Post #1) so we come to this

$$ F_{lift} = \rho * V * g $$ The result should be $F_{lift} = 0,822 N$

Now we need to calculate the force that is pulling the balloon down.Since the baloon is considered to be massless we take no mass into account except the gas within the baloon its self,the Helium.
$$ F_b = \rho * V * g $$ The resoult should be $F_b = 0,113 N$

That means that every baloon provides an upward lift of around 0,709N.Now we can calculate the number of balons we need like this.

$$ n = \frac{Fg}{F_{lift}-F_b} $$ Where Fg is the force that is pulling the child now (m * g)
That should come to n = 484 baloons are needed to make the kid fly.

What do you think,does this make any sence?

 

[haruspex]

Okay so I've taken my sweet time read through some literature my notes and this is what I've got so far.

So if we want to find what the force of lift is,we should note that it is equall to the weight of the compressed fluid so.

$$ F_{lift} = m_{air} * g $$

The mass of air can be calculated as $m = \rho * V$ where V is the volume of the sphere shaped baloon.(the 0,065m^3 I've calculated in Post #1) so we come to this

$$ F_{lift} = \rho * V * g $$ The result should be $F_{lift} = 0,822 N$

Now we need to calculate the force that is pulling the balloon down.Since the baloon is considered to be massless we take no mass into account except the gas within the baloon its self,the Helium.
$$ F_b = \rho * V * g $$ The resoult should be $F_b = 0,113 N$

That means that every baloon provides an upward lift of around 0,709N.Now we can calculate the number of balons we need like this.

$$ n = \frac{Fg}{F_{lift}-F_b} $$ Where Fg is the force that is pulling the child now (m * g)
That should come to n = 484 baloons are needed to make the kid fly.

What do you think,does this make any sence?

Yes, that's the way, and the answer looks about right.
Note that the average molecular weight of helium (i.e. of one atom) is about 4 and that of air about 29, so at the same temperature and pressure the weights would be in a ratio of about 7:1, as you found.
However, in practice, the pressure in the balloon would be a little higher, bringing the ratio down a bit.

I strongly recommend working entirely symbolically until the final step, rather than plugging in numbers straight away. It has many advantages. In the present case, g would have cancelled, eliminating some unnecessary arithmetic.

 

[Orodruin]

eliminating some unnecessary arithmetic.

... not to mention the typical rounding errors that follow that arithmetic ...

To summarise, I would simplify the expression to
$$
n = \frac{m}{V(\rho_{\rm Air} - \rho_{\rm He})}
$$
before inserting numbers.

 

[arhzz]

That is very true,a habit I should get used to.Luckily my professor isn't too strict with the rounding etc,he values approach more than the actual numbers.But thank you for your help as always!