1. An oil bath maintained at 50.5°C loses heat to its surroundings at the rate of 4.68 kJ/min. Its temperature is maintained by an electrically heated coil with a resistance of 60 S2 operated from a 110 V line. A thermoregulator switches the current on and off What fraction of the time will the current be turned on?
2. Calculate the work done on a 2500. kg car in accelerating it from rest to a speed of 100. km/hr. (Neglect friction.)
3. A lead bullet is fired at a wooden plank. At what speed must it travel to melt on impact, if its initial temperature is 25°C and the heating of the plank is neglected? Look up the properties of lead needed.
4. Considering U as a function of any two of the variables P, V, and T, prove that
5. Using the definition H = U + PV and, when necessary, obtaining conversion relationships by considering H (or U) as a function of any two of the variable P, V and T, derive the following relationships:
6. The area a of a rectangle can be considered to be a function of the breadth b and the length 1 so that a = bl . The variables b and 1 are considered to be the independent variables while a is the dependent variable. Other possible dependent variables are the perimeter p and the diagonal d .
A. Derive expressions for the following partial derivatives in terms of b and 1 o r calculate numerical answers.
B. Derive suitable conversion expressions in terms of the partial derivatives given in (A) for each of the following derivatives. Evaluate the results in terms of b and 1 . (Do not substitute the equation for p or d into that for a .)
7. Consider the differential dz = (x2 + y2)dx + 2xydy
i) Show whether or not the differential is exact.
ii) Evaluate Az in going from point (x=0,y=0) to point (x=1,y=1) on a path where y = x2. Compare this to Az between the same points on a path where x = y .
iii) Evaluate Az in going from point (x=0,y=0) to point (x=1,y=1) on a path where y = 0 from point (0,0) to (1,0) and then x = 1 from point (1,0) to (1,1)
8. Repeat the three steps of problem 7 for the differential dz = xydx + xydy