How to Find Root Mean Square Velocity in a Vibrating Box with Steel Balls?

[kpou]

Homework Statement

Thermal Dynamics question, gases?
So I have this box with lengths 20cm on each side. There are 100 balls inside of it with diameter 5mm each. The density in the box is 7.8 g/cm3. The bottom of the box vibrates so the balls bounce around. The top of the box has a movable piston of mass 1kg. What is the root mean square velo of the steel balls if the top of the box is in dynamic equilibrium with the gas of steel balls? Ignore gravity for motion of the balls.

Homework Equations

Equations
pV=NkT,
p=m<v^2>N/V=m(2/3)(U/mN)(N/V)=(2/3)(U/V)
pV=2/3U
(p+a(n/v)^2)((V/n)-b)=RT
U=N<K>=1/2Nm<v^2>

The Attempt at a Solution

What I know is there is const V, const N
And what I am basically stuck on is how can I find pressure without temperature? Or vise versa? I have a feeling the answer might be lying in the statement of the top of the box being in dynamic equilibrium to the gas of steel balls.
Also, what does the mass of the movable piston have to do with this?
All the work I've been doing is likely just garble working with the knowns. I haven't found a formula with <v^2> that works with what I can see.

I feel like if I knew how dynamic equilibrium fit into this it would make this doable. And maybe what the 1kg piston has to do with it as well.

Not being able to find T or p is starting to get to me (sad face)

 

[Redbelly98]

Looks like an interesting -- and challenging -- problem. Do you have a figure along with the problem statement?

Since there is a moveable piston, it seems to me that the volume is not fixed. Also, the weight of the 1 kg mass must be balanced by the "gas" pressure, whatever that is.

If you know the area of the 1 kg piston (is it the entire 20x20 cm^2 of the top of the box?), then you can figure out what the pressure is. Hint: the pressure pushes upward on the 1 kg. Acting down on the 1 kg are the force of gravity and the pressure of the atmosphere.

The density is another clue, since it relates the mass and volume of the gas.

 

[tomcue]

I would suggest approaching this problem by breaking it down into smaller parts and using the equations you have listed to solve for the unknowns.

Firstly, you can use the ideal gas law (pV = NkT) to find the pressure (p) of the gas of steel balls inside the box. You know the volume (V) and number of balls (N), and you can assume a value for the temperature (T) to solve for p.

Next, you can use the equation U = N<K> = 1/2Nm<v^2> to find the average kinetic energy of the steel balls (U). You know the number of balls (N) and can assume a value for the average velocity (<v^2>) to solve for U.

Now, with the value of U, you can use the equation pV = 2/3U to solve for the unknown variable, which in this case is the average velocity (<v^2>).

The statement about dynamic equilibrium means that the forces acting on the top of the box are balanced, which is why it is not moving. This can help you determine the relationship between the pressure of the gas and the mass of the movable piston.

Overall, the mass of the piston does not directly affect the solution to this problem, but it can be used to understand the relationship between pressure and mass in this system.

I hope this helps guide you towards a solution. Remember to use the equations and assumptions to help you solve for the unknown variables.