The First Law of Thermodynamics — Question 4

Question 4

A moving system needs kinetic and potential energy in its balance

A closed 2.00 kg package receives 100 J of heat and 200 J of total external work input. Its center-of-mass speed increases from 3.00 to 7.00 m/s and it rises 5.00 m. Use $\Delta U+\Delta K+\Delta E_g=Q+W_{\mathrm{in}}$ with $g=9.80\,\mathrm{m/s^2}$, treating gravity through $E_g$. The stated work input excludes gravity.

1. Calculate $\Delta K$, $\Delta E_g$ and $\Delta U$.

2. How much additional heat would have to leave during the same motion and work input to make $\Delta U=0$?

Question 4 - Solution

A moving system needs kinetic and potential energy in its balance

(a) Method and calculation

$$\Delta K=\tfrac12(2)(7^2-3^2)=40.0\,\mathrm J,\quad\Delta E_g=2(9.80)(5)=98.0\,\mathrm J.$$$$\Delta U=100+200-40-98=162\,\mathrm J.$$

(b) Interpretation and further calculation

$$0=Q_{\mathrm{required}}+200-40-98\ \Rightarrow\ Q_{\mathrm{required}}=-62.0\,\mathrm J.$$$$Q_{\mathrm{additional}}=Q_{\mathrm{required}}-100=-162\,\mathrm J.$$Relative to the stated 100 J input, an additional 162 J must leave, leaving 62 J net outward heat.

Physical check

Counting all supplied energy as internal energy would incorrectly include the package's bulk motion and elevation.