Trace the curve \(r = a(\cos\theta + \cos 2\theta)\), and shew that the curve crosses itself at the points \((a/\sqrt{2}, \pm \frac{1}{4}\pi)\). Prove that the area of that portion of the largest loop that is not common to the other loops is \(\sqrt{2}a^2\).
Explain in what sense the Kelvin scale of temperature is ``absolute.'' How is it possible to test the agreement between the Kelvin absolute scale and the scale of the Hydrogen thermometer?
Explain the principle of the ``Throttling Calorimeter'' for measuring the dryness of steam, and why this can only be used to measure dryness values between about \(\cdot 95\) and 1.0. Dry sat. steam at pressure 100 lbs./sq. in. abs. is throttled at the stop valve of an engine to 70 lbs./sq. in., what is the temp. of the steam as it enters the engine?
A cylinder of compressed carbon dioxide contains 2.1 lbs. of gas at pressure 120 lbs./sq. in. and temperature 15° C. The cylinder may only be subjected to an internal pressure of 350 lbs./sq. in. and the temperature is liable to rise to 30° C. What further weight of CO\(_2\) would it be safe to add to the contents of the cylinder?
Explain carefully what you understand by `reversibility' as applied to a heat engine. Why, and in what respects, are actual engines irreversible? Compare a steam engine and a gas engine from the point of view of reversibility.
A refrigerating machine has been called a ``Heat-pump''; illustrate the meaning of this by describing the action of the Bell-Coleman refrigerating machine. Obtain an expression for the ``coefficient of performance'' of the cycle, and point out the conditions necessary for a high coefficient. If such a refrigerator is driven by a direct acting Carnot engine taking in heat at \(\tau_1\) and rejecting heat at \(\tau_0\), show that the maximum amount of heat which can be taken from the cold chamber at \(\tau_2\) and delivered to the cooling coils at \(\tau_0\), is \[ \frac{\tau_2(\tau_1-\tau_0)}{\tau_1(\tau_0-\tau_2)} \] thermal units for each thermal unit taken in by the Carnot engine at \(\tau_1\).