Balloon Vacuum: Pressure Calculation w/ R0, σ

[charlie05]

Homework Statement

Balloon in vacuum, radius R0 = 5,0 cm, surface tension σ = 25 N/m.

a/ the pressure in the balloon p in,0 by radius R0………. pin,0 = ?

b/ relation to the pressure inside the balloon pin, depending on its radius R…?

c/relation to the pressure inside the balloon pin , depending on the pressure outside the balloon pout and on its radius R, surface tension is σ…….?

Homework Equations

a/ pin,0=4σ/R0 = 4.25/0,05 = 2kPa

b/ pin = 4σ/R

c/ pin = pout + 4σ/R

The Attempt at a Solution

 

[mfb]

Looks right.

 

[charlie05]

in B/ I have to use pressure p in 0...

p in = 4σ/R ...R = R0 + ΔR = R0 + ( R-R0)...p in = 4σ/R0 + ( R-R0)

 

[mfb]

If you add proper brackets to the denominators, you get the same result as in post 1.

 

[charlie05]

so:?

p in = p in 0 + 4σ/ ( R- R0)

 

[mfb]

That looks wrong, no matter how I interpret the notation, and I don't see how you got it. In particular, it is undefined for the easiest case of R=R₀.

Why do you want to change a correct answer?

 

[charlie05]

Because I have to find dependence p in on p in 0 : Find the equation for the pressure p in inside the balloon, depending on the radius R, where
at radius R 0 is pressure p in 0

 

[mfb]

That's not what you write in post 1.

b/ relation to the pressure inside the balloon pin, depending on its radius R…?

 

[charlie05]

Yes, it's true, I did not read the assignment to the end :-( I am sorrry...

 

[haruspex]

Because I have to find dependence p in on p in 0 : Find the equation for the pressure p in inside the balloon, depending on the radius R, where
at radius R 0 is pressure p in 0

Your notation is hard to interpret, but I think you are saying:
With zero external pressure, the radius is R₀ and the internal pressure is p_in0.
With some unknown external pressure the radius is R; what is the internal pressure, p_in, now?

If that is right, write out the equation for each of the two circumstances. Please use subscripts, as I have, to make the notation clearer. Use the X₂ button above the text entry box.

 

[charlie05]

b/ Find the equation for the pressure p _in inside the balloon, depending on the radius R, where
at radius R ₀ is pressure p _{in 0}

 

[haruspex]

b/ Find the equation for the pressure p _in inside the balloon, depending on the radius R, where
at radius R ₀ is pressure p _{in 0}

Yes, that agrees with my interpretation. So what two equations can you write, one for the zero external pressure case and one for the general case?

 

[charlie05]

in vacuum : ...p_in = 4σ/R...for R₀ ...p_in0 = 4σ/R₀
p _in = 4σ/R ...R = R₀ + ΔR = R₀ + ( R-R₀)...p _in = 4σ/ (R₀ + ( R-R₀))
- But here now I can not express p_in0

for the general case: p_in = p_out + 4σ/R

 

[haruspex]

in vacuum : ...p_in = 4σ/R...for R₀ ...p_in0 = 4σ/R₀
p _in = 4σ/R ...R = R₀ + ΔR = R₀ + ( R-R₀)...p _in = 4σ/ (R₀ + ( R-R₀))
- But here now I can not express p_in0

for the general case: p_in = p_out + 4σ/R

It's all the same balloon and contents. What does not vary as the external pressure changes?

 

[charlie05]

surface tension?

 

[haruspex]

surface tension?

What else?

 

[charlie05]

the amount of air in the balloon ?

 

[haruspex]

the amount of air in the balloon ?

Right. What law can you use?

 

[charlie05]

p₁V₁ =p₂V₂...?

 

[haruspex]

p₁V₁ =p₂V₂...?

Yes. (You can assume the temperature is constant.)

 

[charlie05]

p_in0V₀=p_inV
p_in = p_in0*V₀ / V = p_in0 * (V₀ / V ) = p_in0 * (R₀³ / R³ ) ...?

 

[haruspex]

p_in0V₀=p_inV
p_in = p_in0*V₀ / V = p_in0 * (V₀ / V ) = p_in0 * (R₀³ / R³ ) ...?

Yes.

 

[charlie05]

super..thanks very much :-)

 

[haruspex]

super..thanks very much :-)

Ok, but what is your answer to the question, i.e. as an equation for p_in in terms of p_out, R and σ?

 

[charlie05]

oh yes...p_in = p_out + 4σ/R...?

 

[haruspex]

oh yes...p_in = p_out + 4σ/R...?

Sorry, my mistake... I meant the question posed in your post #7, where you need to find p_in as a function of R, R₀ and p_{in 0}.

 

[charlie05]

once again thank you very much for your help...