Calculating Motion of a Test Balloon on a Methane Planet

[robax25]

Homework Statement

A planet has a gaseous atmosphere with a density of 0.1 g/cm³. The planet itself
completely consists of liquid methane with a density of 0.4 g/cm³ and has a radius of
3800 km. A test balloon is dropped onto this planet, which consists of a helium spherical
balloon (1 m³, mass 15 kg) and a box of test equipment (10 cm x 10 cm x 10 cm, mass 4 kg)
attached by a 10 m long rope.
Consider the case when the box is in the liquid methane and the balloon is at the surface.
Calculate the motion of the box and draw x(t), v(t) and a(t) diagrams until the boxes reaches
the surface, quantitatively

Homework Equations

y=-0.5at²

G=6.67 *10^(-11) m³/kgs²
g(for the planet)=mG/r²

The Attempt at a Solution

g=9,81 kgm/s²

I need value of g for the planet.
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[BvU]

I need value of g for the planet.

Well, in your own words:

g(for the planet)=mG/r²

So what are you missing ?

 

[robax25]

mass of the planet

 

[BvU]

You are given the density and the radius. Planets are spherical ...

 

[robax25]

i am not sure how to proceed

 

[BvU]

When unsure how to proceed, make a diagram

 

[robax25]

I get it how to do it. can you tell me please which density should I consider?Should I sum up both density.?.

 

[BvU]

For the planet you take the density of the planet

 

[robax25]

you mean 0,1g/cm³

 

[BvU]

The planet itself completely consists of liquid methane with a density of 0.4 g/cm³

 

[robax25]

Then I get the value of g=0.414 m/s²

 

[BvU]

I get 0.425

 

[robax25]

yes.0 425

 

[robax25]

i just draw Displacement vs time graph

 

[robax25]

can you tell me please whether it is right or wrong

 

[BvU]

What am I supposed to see ? Perhaps you want to work it out a little more ?
And: are you sure you have the right scenario in mind ?

Consider the case when the box is in the liquid methane and the balloon is at the surface.
Calculate the motion of the box and draw x(t), v(t) and a(t) diagrams until the boxes reaches the surface, quantitatively

 

[robax25]

yes, he just throws it from 10m. Total mass=19kg.

 

[BvU]

Balloons have a tendency to go up. Not this one ?

 

[robax25]

but if you attach Ballons and box together and throw them from 10m on a Liquid surface. You will see the same thing that he explains.

 

[robax25]

If I miss something, please explain it correctly.

 

[BvU]

I would prefer it if you check your own assumptions -- do you really think the balloon is going down ?

 

[robax25]

balloon will not go down. However, If you put heavy thing attached with balloon, they will go down.

 

[robax25]

I think i should only consider the mass of box

 

[BvU]

Box is 4 kg, balloon 15.

What did you use the density of the atmosphere of that planet for ? Or was it given just to distract you ?

 

[robax25]

if you consider air resistance, then, atmosphere of the planet is necessary.If you neglect air resistance, then, it is not necessary.I would like to consider. If i consider, then, the value of acceleration would change.

 

[BvU]

What is the mass of 1 m3 of this planets atmosphere at the surface ?

 

[robax25]

100kg as ρ=0.1 g/cm³ if i multiply with 1000 then the value is 100kg/m³

 

[BvU]

Good. So according to Archimedes, the planet pulls harder on 1m3 of atmosphere than on 1 m3 of helium balloon with a mass of 4 kg hanging from it, isn't it ? Make a free body diagram and see what the resulting force is.

 

[robax25]

resulting force =-40.8 N as g=0.425 m/s² so should the balloon weight be negligible? Yes balloon force acts up, it always blows up.

 

[BvU]

'Blowing up' is not how I would describe it .

I wonder if the exercise composer really intended to make things so complicated with this scenario:

Consider the case when the box is in the liquid methane and the balloon is at the surface

We are supposed to look at the box, which initially is at -10 m.

I don't see how you can find an upward net force of 40.8 N. Can you show the details ?

Make a free body diagram and see what the resulting force is

Then you can decide if the mass of the balloon can be ignored (not). Idem mass of box (not). And the upward force on the submerged box (probably).

Nothing is said about the viscosity of the liquid or gaseous methane, so I propose to ignore that. The exercise is already tough enough.

 

[robax25]

how can I get motion of the box?

 

[haruspex]

how can I get motion of the box?

The starting state is that the box is 10m down in the methane and the balloon is at (on) the surface. You can suppose the balloon is not immersed at all in the methane, since it will not be for long.
What are the forces on the balloon? What are the forces on the box?
(You may be tempted to consider the balloon, rope and box as a single system and only consider the external forces on that, but that is not safe. Do you see why?)

 

[robax25]

the box has force F(box)=mg and balloon is full of Helium. It goes up.The atmospheric density changes the motion of the box.I need to use Archimedes formula

 

[robax25]

I can use this formula to get Drag force=Cd ρ v² *A/2 and we don't know the value of Cd, we can skip it? does It ok?

 

[haruspex]

need to use Archimedes formula

Quite so. What is the total buoyant force?

I can use this formula to get Drag force=Cd ρ v² *A/2 and we don't know the value of Cd, we can skip it? does It ok?

I wouldn't worry about drag. There is nowhere near enough information to quantify that.

 

[robax25]

total Buoyant force= m(Ballon)g-m(Box)g=15kg*0.425m/s² - 4kg*0.425m/s²=4.675 N

 

[haruspex]

total Buoyant force= m(Ballon)g-m(Box)g=15kg*0.425m/s² - 4kg*0.425m/s²=4.675 N

The buoyant force has nothing to do with the masses of the objects.
Quote Archimedes' principle.

 

[robax25]

The box is immersed in methane liquid . So Buoyant force on the immersed box is 0.17 N and Weight of the box is = 1.7 N

 

[haruspex]

Buoyant force on the immersed box is 0.17 N

No. How are you calculating that?
Please state Archimedes' Principle.

 

[robax25]

I uploaded the pdf file. Archimedes principle state that the upward force acts on an object is equal to weight of the fluid that the body displaces

 

[haruspex]

I uploaded the pdf file. Archimedes principle state that the upward force acts on an object is equal to weight of the fluid that the body displaces

Sorry, my mistake. (Earlier you had a term that equated to 1.7N, and did not notice you were now writing 0.17N.) But you could have got there a bit more simply using Archimedes' principle.

So now, what is the buoyant force on the balloon? You will have to use Archimedes for that.

 

[robax25]

Buoyant force for BallonFb=170N

 

[haruspex]

Buoyant force for BallonFb=170N

The balloon is not immersed in the liquid methane. Take it as sitting on or a bit above the surface.

 

[robax25]

but Buoyant force does not depend on either it is submerged or immersed. It will be same if object is submerged or immersed. Buoyant force is telling you either object will float or not. In this case, Buoyant force is greater than the Force of gravity of the Ballon.

 

[haruspex]

but Buoyant force does not depend on either it is submerged or immersed. It will be same if object is submerged or immersed. Buoyant force is telling you either object will float or not. In this case, Buoyant force is greater than the Force of gravity of the Ballon.

Whether you want to call it immersed or submerged, the balloon is not in the liquid, so does not get any buoyant force from the liquid. What is exerting a buoyant force on the balloon, and what is its density?

 

[robax25]

I get it now. Buoyancy force is exerted by atmosphere of the planet which has density 100kg/m³. so Fb= 100kg*0.425 m/s² =42.5 N

 

[haruspex]

I get it now. Buoyancy force is exerted by atmosphere of the planet which has density 100kg/m³. so Fb= 100kg*0.425 m/s² =42.5 N

Yes.

 

[robax25]

how to proceed now?I need to draw st, vt, at graph?

 

[haruspex]

how to proceed now?I need to draw st, vt, at graph?

Yes, but first you need to find the actual equation for the motion.

 

[robax25]

which kind of equation? can you give me some hints please.