Balloon in train

1. Nov 7, 2016

Hamal_Arietis

1. The problem statement, all variables and given/known data
As show in the figure below, balloon A (containing gas of density $ρ_A$) and balloon B (containing gas of density $ρ_B$) are each suspended by string from the ceiling of a train at rest. Balloon C (containing gas of density $ρ_C$ is attached to the train floor by string, and floats above the floor. The density of the air is $ρ_o$, where $ρ_C<ρ_o<ρ_B<ρ_A$. The balloon's mass, the string's mass at the movement of air inside the train are negligible.

The train begins moving with uniform accleration in a horzizontal direction (to the right in the figure). A, B, S come to rest with respect to the train and form, respectively, angles $\theta_A;\theta_B;\theta_C$ with the vertical. Choose the best represents the condition inside the train at this time.

2. Relevant equations
F=-ma
3. The attempt at a solution

$$tan \theta_A=\frac{g\rho_A V-g\rho_oV}{V\rho_Aa}$$
$$tan \theta_B=\frac{g\rho_B V-g\rho_oV}{V\rho_Ba}$$
Because $\dfrac{\rho_A-\rho_o}{\rho _A}>\dfrac{\rho_B-\rho_o}{\rho _B}$
and I choose 1. But the answer is 4. Where is my wrong?

Last edited: Nov 7, 2016
2. Nov 7, 2016

jbriggs444

If you are sitting in a chair in the train car, you can consider the train's forward acceleration as equivalent to a rearward pull of a gravity-like force. Together with gravity, the net effect is as if the direction of gravity had tilted.

What does this mean for the direction of buoyancy?

3. Nov 7, 2016

Hamal_Arietis

the g' inside the train is equal $g'=\sqrt{g^2+a^2}$ but they are effected by Achimede force.
The 4 answer is right. But I dont understand why the balloon C is equilibrium.

4. Nov 7, 2016

CWatters

The buoyancy force acts in the same direction as the net acceleration.

You think C should be moving?

5. Nov 7, 2016

TomHart

I think for most people it seems hard to believe that the helium balloon (or whatever gas it is using) would move in the direction that it does under acceleration of the vehicle. And probably the reason it is so counterintuitive is that everything we have seen that moves under acceleration of a vehicle moves in the opposite direction - including the direction our body tends to move, or something sitting loosely on the dashboard or on the floor.

I had wondered about this previously and had the opportunity to perform an experiment one time with a helium balloon in my car. It had a small weight attached to the string and I set it beside me as I was driving. It almost seemed a bit comical when the balloon leaned into the direction of the turn or leaned backward under braking and forward under acceleration - almost as if it was bracing itself for the normal movement that it was expecting.

An apple falls from a tree because it displaces the air that is less dense. Under acceleration, the air in the vehicle moves in the opposite direction of the acceleration because it displaces the less dense helium.

6. Nov 7, 2016

Hamal_Arietis

yes because $\vec{P}+\vec{F}+\vec {F_a}$ isnt equal $\vec{T}$

7. Nov 7, 2016

Hamal_Arietis

Can you use mathematics equations to answer this question?

8. Nov 7, 2016

jbriggs444

Can you define $\vec{P}$, $\vec{F}$, $\vec{F_a}$ and $\vec{T}$ so that we know what physical inequality you are asserting? None of those variable names have been mentioned in the thread up to this point.

9. Nov 7, 2016

Hamal_Arietis

So Balloon C can't be equilibrium

10. Nov 7, 2016

jbriggs444

$\vec{T}$ is the tension in the string pulling down and left.
$\vec{F}$ is the leftward pseudo-force resulting from the use of the accelerating frame.
$\vec{F_a}$ is the upward?! force arising from the pressure gradient in the air
$\vec{P}$ is the downward force from gravity.

Which direction is the pressure gradient in the air?

11. Nov 7, 2016

Hamal_Arietis

Oh, I don't notice that pressure gradient in the air. But how 3 angels equal?

Last edited: Nov 7, 2016
12. Nov 7, 2016

jbriggs444

Realize that the vector sum of the downward force of gravity and the leftward pseudo-force from the train's acceleration produce a result that is exactly like a motionless room that is tilted (and subject to slightly increased gravity).

The air is subject to the same net gravity as everything else. Its pressure gradient lines up accordingly.

Edit: I assume that you understand that "buoyancy", "the Archimedes force" and "the force arising from the pressure gradient" are all descriptions of the exact same thing.

13. Nov 7, 2016

Hamal_Arietis

Ok, now I find the gradien of air
$$[p (x)- p (x + dx)] S = ma = a \rho_0 dx$$
So $\Delta p = -a \rho_0 \Delta x \Rightarrow \frac{\Delta p}{\Delta x}=-a\rho_0=const$
So 3 balloon are effected by equal force
The force by gradien of C balloon is larger than the force F=ma of its
I think it wrong.
The gravity in train is:
$$g'=\sqrt{g_0^2+a^2}=\sqrt{(g-\frac{\rho_og}{\rho_i})^2+a^2}$$
So g' is denpendent with $\frac{\rho_o}{\rho_i}$
Why 3 angles are equal?

Last edited: Nov 7, 2016
14. Nov 7, 2016

Hamal_Arietis

SO I have the figure

15. Nov 7, 2016

jbriggs444

Define your variables before you put them into formulas.
Explain what physical principle motivates the formulas that you write down.
A verbal description would make it much easier to decipher the chicken scratchings above.

16. Nov 7, 2016

Hamal_Arietis

Sorry.
Because English isn't my national language that showing my idea in science language is very difficult. But I can understand your idea. And I tried showing my idea
#13. Firstly I find the gradien pressure force effect to 3 balloon And prove that C balloon doesn't move.
Secondly I show that g' (is virtual gravity acceleration) is dependent with $\dfrac{\rho_0}{\rho_i}$ (i=A,B,C) so the 3 angles can't equal.
But I don't inderstand why angle A is equal angle B.

17. Nov 7, 2016

CWatters

Have you considered what happens if (instead of accelerating the container) you simply rotate the container so that the net acceleration (just due to gravity) points in a different direction...

18. Nov 7, 2016

Hamal_Arietis

But the train moves horizontally with acceleration $\vec{a}$ and 3 balloon is effected by gravity and Achimede force

19. Nov 7, 2016

Hamal_Arietis

YEs, I understand what you say
It mean?

20. Nov 7, 2016

CWatters

Correct.

Gravity and the horizontal acceleration add together (vector add) to make a net acceleration.

The "Archimedes force" (buoyancy force) always acts in the opposite direction to the net acceleration.