A Closer Look at Heat and Work — Question 6

Question 6

A spring makes piston pressure grow linearly with volume

A frictionless piston with area $.0200\,\mathrm{m^2}$ begins against baseline pressure $p_0=100\,\mathrm{kPa}$. A spring is initially uncompressed and has stiffness $k=2000\,\mathrm{N/m}$. Quasistatic outward motion compresses the spring by $x$. Let final $x=.300\,\mathrm m$.

1. Derive $p(V)$ and calculate total gas work.

2. Find the portion stored in the spring and the final pressure. Verify the work using average pressure.

Question 6 - Solution

A spring makes piston pressure grow linearly with volume

(a) Method and calculation

$$V-V_i=Ax,\quad p=p_0+\frac{kx}{A}=p_0+\frac{k}{A^2}(V-V_i).$$$$W=\int_0^{.300}(p_0A+kx)dx=p_0A(.300)+\frac12k(.300)^2=690\,\mathrm J.$$

(b) Interpretation and further calculation

$$E_{\mathrm{spring}}=\tfrac12kx^2=90.0\,\mathrm J,\quad p_f=100000+2000(.300)/.0200=130000\,\mathrm{Pa}.$$$$W=\frac{p_i+p_f}{2}\Delta V=115000(.00600)=690\,\mathrm J.$$The pressure-average shortcut is exact because this pressure law is linear in volume.

Physical check

The spring contribution is one-half the final spring force times displacement, not the full final force times displacement.