Pressure vs length vs optimal speed vs volume

In summary, to calculate the optimal cylinder length for the highest speed of the round object, you need to consider the mass of the object, the force exerted on the piston, and the temperature and pressure of the air inside the cylinder. By using the equation of motion and the ideal gas law, you can determine the optimal cylinder length for maximum speed.
  • #1
Tux
1
0
First off, let me say that I have spent hours trying to figure out how to calculate what I want here, but to no avail.

I guess I should go into this problem's details...

What I have here are two cylinders. I balanced the volumes of these two cylinders before experimentation for the sake of simplicity.

The first cylinder has a piston in it that exerts a certain pressure to the gas inside the cylinder, therefore changing the volume of the gas.

This pressure builds up via a stopper that happens to be a round object at the opening of the second cylinder. This object has a certain force that has to be applied to it in order for it to move (it is held into place by a small rubber stopper). Th point of this object is to be propelled by the pressure from the piston out of the second cylinder.

Now, the whole point here is that I want this round object to leave the second cylinder at the highest speed possible which means that it needs to conserve pressure behind it.


Here's my data and work so far:

Cylinder 1:

Diameter of cylinder 1 = 15.88 mm
PI = 3.14
Length of cylinder 1 = 98.73 mm
Volume of cylinder 1 = 4922.99 mm ^ 3

Volume of cylinder 1's lip/nipple that fits into cylinder 2:
506.71 mm ^ 3

Total volume of cylinder 1 (including nipple):
5429.7mm ^ 3


Cylinder 2:

Diameter of cylinder 2 = 5.98 mm
PI = 3.14
Length of cylinder 2 = 289.16 mm
Volume of cylinder 2:
5429.7mm^3


Alright now, this is nice and balanced considering room pressure and temperature but what if I decide to exert 2.0 joules of force on the piston and compress the air? How would I calculate the optimal cylinder length for speed of the round object?

This is not homework, this is a real world example that I need to figure out really soon. I will appreciate all of your help.

Thanks!


Example is attached.
 

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  • #2
In order to calculate the optimal cylinder length for the highest speed of the round object, you need to consider the following variables:1. The mass of the round object2. The area of the opening of the second cylinder3. The force exerted on the piston4. The pressure of the air inside the cylinder5. The volume of the cylinder6. The temperature of the air inside the cylinderThese are all important factors that will determine the optimal cylinder length for maximum speed.To calculate the optimal cylinder length, you will have to use the equation of motion. This equation states that the final velocity of an object is equal to its initial velocity plus the acceleration (due to force) multiplied by the time interval. This equation can be expressed as follows:Vf = Vi + (F/m)*tWhere:Vf = Final velocityVi = Initial velocityF = Force exerted on the pistonm = Mass of the round objectt = Time intervalYou can then rearrange the equation to solve for t:t = (Vf-Vi)/(F/m)Once you have calculated the time interval, you can then use the equation of displacement to calculate the optimal cylinder length for maximum speed:d = (Vf+Vi)*t/2Where:d = Optimal cylinder lengthVf = Final velocityVi = Initial velocityt = Time intervalOnce you have the optimal cylinder length, you can then use this value to calculate the pressure of the air inside the cylinder using the ideal gas law:PV = nRTWhere:P = Pressure of the air inside the cylinderV = Volume of the cylindern = Number of moles of air in the cylinderR = Universal gas constantT = Temperature of the air inside the cylinderCombining the two equations, you can then calculate the optimal cylinder length for maximum speed based on the force exerted on the piston, the mass of the round object, and the temperature and pressure of the air
 
  • #3


I understand the complexity of your problem and the importance of finding a solution quickly. In order to calculate the optimal cylinder length for speed of the round object, we would need to take into account several factors such as the gas law, the compressibility of the gas, and the friction between the round object and the walls of the cylinder.

Firstly, we need to determine the gas law that applies to your experiment. Is the gas inside the cylinder an ideal gas (following the ideal gas law) or a real gas (following the van der Waals equation)? This will affect the calculation of the pressure and volume relationship.

Secondly, we need to consider the compressibility of the gas. As pressure is applied to the gas, it will compress and its density will increase. This will affect the speed of the round object as it moves through the gas.

Lastly, we need to take into account the friction between the round object and the walls of the cylinder. This will depend on the material of the cylinder and the shape and size of the round object. Friction will slow down the round object and decrease its speed.

To calculate the optimal cylinder length, we would need to use a combination of mathematical equations and experimental data. It would be helpful to have a pressure gauge and a speed measuring device to gather accurate data. We would also need to perform multiple trials with different cylinder lengths to determine the optimal length for speed.

I hope this helps guide you in your calculations. Please let me know if you need further assistance or clarification. Best of luck with your experiment!
 

What is pressure?

Pressure is defined as the force per unit area applied perpendicular to the surface of an object. It is typically measured in units of Pascals (Pa) or pounds per square inch (psi).

How does pressure affect length?

Pressure and length are directly proportional. This means that as pressure increases, length also increases. This can be seen in phenomena such as the expansion of a balloon when it is filled with air.

What is optimal speed?

Optimal speed refers to the ideal velocity at which a system or object operates most efficiently. In the context of pressure and length, optimal speed would be the speed at which the system experiences the least amount of pressure and length change.

How does volume change with pressure and length?

Volume is inversely proportional to pressure and length. This means that as pressure and length increase, volume decreases. This can be seen in the compression of gases, where an increase in pressure and decrease in volume occur simultaneously.

What are some practical applications of understanding pressure, length, optimal speed, and volume?

Understanding the relationship between pressure, length, optimal speed, and volume is crucial in various industries such as engineering, physics, and chemistry. It can help in designing efficient systems, predicting the behavior of gases and liquids, and optimizing processes. Some practical applications include designing efficient engines, predicting weather patterns, and developing medical devices.

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