Finding spring constant when gas expands against a spring

[BlackMamba]

Hi there,

I have a problem that I believe I am doing correctly, but my answer proves otherwise. I was hoping someone could take a look and let me know where I'm going wrong.

Here's the problem: A gas fills the right portion of a horizontal cylinder whose radius is 5.10cm. The initial pressure of the gas is 1.01x10^5 Pa. A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is 18.0cm. When the pin is removed and the gas is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.


So here's what I've done. The final equation that I'm going to need to get to is the ole F = kx equation. And since the length doubles it would probably look something like F = k[2(x)]

But alas I'm missing F and I believe x after converting to meters would be 0.180m.

So I have a Pressure and radius. With the radius I found the area using A = (pie)r^2

Now I have an Area.

From there I used the equation P = F/A to find F. And once getting F I went back to my initial equation plugging in F and solving for k. But my answer is not correct. I was almost sure this was how to solve this problem, but I guess not.

Any help would be greatly appreciated.

 

[spacetime]

Energy stored in the spring after it is compressed to a distance x is

$$\frac{1}{2}kx^2$$

This is equal to the work done on it by the gas due to its expansion. So, to calculate the work done by the gas, use the equation for work done by the gas in an isothermal expansion.

$$W=nRTln\frac{v_f}{v_i}$$

where $$nRT=p_iV_i$$
Hope that helps!

spacetime
www.geocities.com/physics_all

 

[wais]

Hi there,

Thanks for reaching out for help with this problem. It looks like you have a good understanding of the concepts involved, but there may be a few small errors in your calculations.

First, when finding the area of the cylinder, make sure to use the radius in meters, not centimeters. So the area would be A = π(0.051m)^2 = 0.0082m^2.

Next, when using the equation P = F/A, make sure to use the pressure in Pascals (Pa), not kilopascals (kPa). So the force would be F = (1.01x10^5 Pa)(0.0082m^2) = 828 N.

Now, when finding the spring constant, you will need to use the equation F = kx, where F is the force you just calculated and x is the change in length of the spring. Since the length of the gas-filled chamber doubles, the change in length would be 0.180m (final length) - 0.090m (initial length) = 0.090m. So the spring constant would be k = F/x = 828 N/0.090m = 9200 N/m.

I hope this helps and clarifies any confusion. Keep up the good work and keep practicing!