Cylinders filled with liquid; placed on a trolley

In summary, we are given two cylinder-shaped vessels connected by a thin horizontal tube on a trolley at rest. The left vessel is initially filled with liquid of density ##\rho## and the total mass of the system is m. After opening the tap, the liquid travels at a speed of v in both vessels. Using conservation of linear momentum, we can find that the speed of the trolley at the moment when the water levels in the vessels are v is given by ##vAL\rho/m##.
  • #1
Saitama
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Homework Statement


Two cylinder shaped vessels of cross section A are fixed vertically to a trolley, which was initially at rest. The two vessels are connected with a thin horizontal tube which is equipped with a tap. The distance between the axes of the vessels is L. The vessel in the left hand side is filled with some liquid of density ##\rho##; and at this time the total mass of the system at rest is m. After opening the tap what is the speed of the cart at the moment when the speed of the water levels in the vessels is v? (Rolling friction, friction in the bearings and air-drag are negligible.)

(Answer: ##vAL\rho /m##)

Homework Equations


The Attempt at a Solution


I think I have to use conservation of linear momentum but I am unsure about how to do that. The motion of fluid in the cylinder confuses me. I need a few hints to get started with.

Thanks!
 

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  • #2
Oh I see, mass transfers from the LHC to the RHC at a rate that depends on the difference in heights - which makes the cart move the other way ... it's like a continuous version of the problem where you sit at one end of a closed box and throw rotten fruit at the other end where they stick.
 
  • #3
The key to the problem, obviously, is finding the momentum of the liquid in the pipe.
 
  • #4
voko said:
The key to the problem, obviously, is finding the momentum of the liquid in the pipe.

And how should I do that? :rolleyes:
 
  • #5
Pranav-Arora said:
And how should I do that? :rolleyes:

Momentum is mass times velocity. Find these.
 
  • #6
voko said:
Momentum is mass times velocity. Find these

The total mass of the system is m. Initially all the water is in the left cylinder. Mass of liquid is ##Ah\rho## where h is the height of the liquid in the left cylinder initially. But I don't think calculating these would help. Also, how do I use ##L## here? :confused:
 
  • #7
Again, you need the momentum in the pipe. It is the mass of the liquid in the pipe multiplied by the velocity of the liquid in the pipe.

What is the mass of the liquid in the pipe?
 
  • #8
voko said:
What is the mass of the liquid in the pipe?
I don't have the cross section for the pipe. :(
 
  • #9
Pranav-Arora said:
I don't have the cross section for the pipe. :(

Assume you do: call it ##a##.
 
  • #10
voko said:
Assume you do: call it ##a##.

The length of the pipe is ##L-2R## where R is the radius of cylinder.

The mass of liquid in the pipe is ##\rho a(L-2R)##. Momentum is ##\rho a(L-2R)v##.
 
  • #11
Pranav-Arora said:
The length of the pipe is ##L-2R## where R is the radius of cylinder.

I believe you are supposed to think that ##L## is much greater than ##R## and ignore the latter. You will have to, anyway, because you do not have enough information to model the behavior of the liquid in the cylinder, and it is not physical to assume it just moves horizontally wall to wall and then goes straight up.

The mass of liquid in the pipe is ##\rho a(L-2R)##. Momentum is ##\rho a(L-2R)v##.

##v## is the velocity in the cylinders, not in the pipe.
 
  • #12
voko said:
##v## is the velocity in the cylinders, not in the pipe.

:tongue:
Sorry about that.

Using the equation of continuity (?),
[tex]Av=av' \Rightarrow v'=Av/a[/tex]
Hence momentum is ##\rho aLv'=\rho ALv##. From conservation of linear momentum, ##\rho ALv=mv_f \Rightarrow v_f=\rho ALv/m##. Looks good?
 
  • #13
This agrees with the answer given.

However, I have a reservation. The velocity ##v'## is with respect to the pipe, which is fixed to the trolley. You are required to find the velocity of the trolley with respect to the ground.
 
  • #14
voko said:
This agrees with the answer given.

However, I have a reservation. The velocity ##v'## is with respect to the pipe, which is fixed to the trolley. You are required to find the velocity of the trolley with respect to the ground.

:confused:

So ##v'=v_L-v_P##? where ##v_L## is velocity of liquid and ##v_P## is velocity of pipe with respect to ground. What should I replace ##v_P## with? :confused:
 
  • #15
##v_p## is what you are supposed to find.
 
  • #16
voko said:
##v_p## is what you are supposed to find.

So what about ##v_L## then? :uhh:
 
  • #17
Obviously, ##v_L = v' + v_P## :)

The mass of the liquid in the pipe has velocity ##v_L##; everything else has velocity ##v_P##,
 
  • #18
voko said:
Obviously, ##v_L = v' + v_P## :)
:tongue:
The mass of the liquid in the pipe has velocity ##v_L##; everything else has velocity ##v_P##,
Conserving linear momentum,
[tex]\rho aL(v'+v_P)=mv_f[/tex]
Is this what I am supposed to solve?
 
Last edited:
  • #19
##v## is the vertical velocity in the cylinders; I do not see how it could possibly be in that conservation law. Besides, ##m## is the mass of the entire system, which includes the mass in the pipe, which is traveling at a different velocity, so that cannot be right, either.
 
  • #20
voko said:
##v## is the vertical velocity in the cylinders; I do not see how it could possibly be in that conservation law.
Woops sorry, I should have used a different symbol.
Besides, ##m## is the mass of the entire system, which includes the mass in the pipe, which is traveling at a different velocity, so that cannot be right, either.

So what am I supposed to do now? :confused:
 
  • #21
Pranav-Arora said:
So what am I supposed to do now? :confused:

Correct your mistakes in the equation for conservation of momentum, I would think.
 
  • #22
voko said:
This agrees with the answer given.

However, I have a reservation. The velocity ##v'## is with respect to the pipe, which is fixed to the trolley. You are required to find the velocity of the trolley with respect to the ground.
I don't see the difficulty. Let m' be the mass of the water in the pipe. If it has velocity v' wrt the trolley, and the trolley has velocity u wrt ground, we have m'(v'+u) + (m-m')u = 0, so m'v' + mu = 0.
 
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  • #23
haruspex said:
I don't see the difficulty. Let m' be the mass of the water in the pipe. If it has velocity v' wrt the trolley, and the trolley has velocity u wrt ground, we have m'(v'+u) + (m-m')u = 0, so m'v' + mu = 0.

The difficulty was that Pranav was having difficulty rolling out this argument, which I hoped he would in the end, but you ruined the party :)
 
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  • #24
voko said:
The difficulty was that Pranav was having difficulty rolling out this argument, which I hoped he would in the end, but you ruined the party :)

Sorry, I read it as if you had found a fault in the given answer.
 
  • #25
haruspex said:
I don't see the difficulty. Let m' be the mass of the water in the pipe. If it has velocity v' wrt the trolley, and the trolley has velocity u wrt ground, we have m'(v'+u) + (m-m')u = 0, so m'v' + mu = 0.

Thank you voko and haruspex!
 

1. What is the purpose of placing cylinders filled with liquid on a trolley?

The purpose of placing cylinders filled with liquid on a trolley is to safely transport and store the liquid. The trolley provides a stable and mobile platform for the cylinders, making it easier to move them from one location to another.

2. How much liquid can a cylinder on a trolley hold?

The amount of liquid a cylinder on a trolley can hold depends on the size and type of the cylinder. Different cylinders have different capacities, ranging from a few liters to several hundred liters.

3. What types of liquids are typically stored in cylinders on a trolley?

Cylinders on a trolley are commonly used to store and transport compressed gases, such as oxygen, nitrogen, and helium. They can also be used for storing and transporting other liquids, such as liquefied petroleum gas (LPG) and liquid nitrogen.

4. Are there any safety precautions to consider when using cylinders filled with liquid on a trolley?

Yes, there are several safety precautions to consider when using cylinders filled with liquid on a trolley. These include properly securing the cylinders on the trolley, following proper handling and storage procedures, and ensuring the trolley is in good condition and can support the weight of the cylinders.

5. Can the liquid in the cylinders on a trolley leak or spill?

Yes, if the cylinders are not properly sealed or secured, the liquid inside can leak or spill. It is important to regularly check for any leaks or damage to the cylinders and to follow proper handling and storage procedures to prevent accidents.

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