Problem 1:
A polar bear partially supports herself by pulling part of her body out of the water onto a rectangular slab of ice of volume 10 m3. The ice sinks down so that only half of what was once exposed now is exposed, and the bear has 60% of her volume (and weight) out of the water. Assume that the bear and the water have mass density 1 g/cm3 and ice has mass density 0.9 g/cm3. Estimate the bear’s mass.
Problem 2:
Pure water can be super-cooled at standard atmospheric pressure to below its normal freezing point of 0 °C. Assume that a mass of water has been cooled as a liquid to –5 °C, and a small (negligible mass) crystal of ice is introduced to act as a “seed” or starting point of crystallization. If the subsequent change of state occurs adiabatically and at constant pressure, what fraction of the system solidifies? Assume the latent heat of fusion of the water is 80 kcal/kg and that the specific heat of water is 1 kcal/(kg oC).
Problem 3:
A thin vertical uniform wooden rod is pivoted at the top and immersed in water as shown.
The pivot point is slowly lowered. At a certain moment the rod begins to deflect from the vertical. What fraction of the rod is still in the water at that moment if the density of the rod is one-half of the density of water?
Problem 4:
A vertical open glass tube of length h is half-submerged in mercury. The top end of the tube is then closed and the tube is slowly pulled out until the bottom of the tube is barely submerged in the mercury. What is the length of the mercury column remaining in the tube? The atmospheric pressure corresponds to the pressure of a column of mercury of height H. Assume the temperature is constant.
Problem 5:
The figure below shows a
maximally efficient Carnot cycle in the pressure-volume plane for a heat engine
operating between two heat reservoirs to perform work.
(a) For each label 1 through 4 identify the process, say whether work is done
by the working fluid or on it and whether heat is added to the working fluid or
extracted from it.
(b) Make a diagram showing the same cycle in the temperature-entropy plane.
Label the parts of your diagram that correspond to the parts labeled in the P-V
diagram and put arrows on each segment indicating the direction it is traversed.
Problem 6:
A nursery uses natural gas heating to keep the greenhouses at 30 oC all year. An engineer points out that the water at the bottom of a nearby lake is at a constant temperature of 5 oC, and that he can build an ideal heat pump that will work at maximum possible efficiency to pump heat from this lake water into the greenhouses. He claims that the nursery will come out ahead with his system, even though it uses electricity instead of natural gas at three times the cost per Joule. Is he right? Neglecting capital and maintenance costs by what factor would their energy bill change?
Problem 7:
In a Wilson cloud chamber at a temperature of 20 degrees C, particle tracks are made visible by causing condensation on ions by an approximately reversible adiabatic expansion of the volume in the ratio
final volume/initial volume = 1.375.
The ratio of the specific heats of the gas at constant pressure and at constant volume is CP/CV = 1.41. Estimate the gas temperature after the expansion.