Physics homework #10

1. An aluminum cylinder 10 cm long, with a cross-sectional area of 20 cm

2

, is to be used as a spacer

between two steel walls. At 17.2

◦

C it just slips in between the walls. When it warms to 22.3

◦

C,

calculate the stress in the cylinder and the total force it exerts on each wall, assuming that the walls
are perfectly rigid and a constant distance apart.

2. You are designing an electronic circuit made of 23 g of silicon. The electric current through it adds

energy at the rate of 7.4 mW. If your design does not allow any heat transfer out of the element, at
what rate does its temperature increase? The specific heat of silicon is 705 J/kg·K.

3. A heavy copper pot of mass 2 kg (including the copper lid) is at a temperature of 150

◦

C. You pour

0.10 kg of water at 25

◦

C into the pot, then quickly close the lid of the pot so that no steam can

escape. Find the final temperature of the pot and its contents, and determine the phase (liquid or
gas) of the water. Assume that no heat is lost to the surroundings.

4. A typical tank used for scuba diving has a volume of 11.0 l and a gauge pressure, when full, of

2.10 × 10

7

Pa. The “empty” tank contains 11.0 l of air at 21

◦

C and 1 atm. When the tank is filled

with hot air from a compressor, the temperature is 42

◦

C and the gauge pressure is 2.11×10

7

Pa. What

mass of air was added? (Air is a mixture of gases: 78% nitrogen, 21% oxygen, 1% miscellaneous; its
average molar mass is 28.8 g/mol.)

5. Show that the variation of atmospheric pressure with elevation in the earth’s atmosphere is

p(y) = p

0

e

−M gy/RT

. Assume that the temperature is 0

◦

C at all elevations. Ignore the variation of g

with elevation.

6. The compression ratio of a diesel engine is 15 to 1. (a) If the initial pressure is 1.01 × 10

5

Pa and

the initial temperature is 27

◦

C, find the final pressure and the temperature after compression. Treat

air as an ideal gas wit γ = 1.4. (b) How much work does the gas do during the compression if the
initial volume of the cylinder is 1 l. Assume that C

V

for air is 20.8 J/mol·K.

7. Find the efficiency of a Carnot engine which is in contact hot reservoir having temperature T

H

and

cold reservoir at T

C

.

8. Suppose 0.2 mol of an ideal diatomic gas (γ = 1.4) undergoes a Carnot cycle with temperatures

227

◦

C and 27

◦

C. The initial pressure is p

a

= 10

6

Pa, and during the isothermal expansion at the

higher temperature the volume doubles. (a) Find Q, W , and ∆U for each step and for the entire
cycle. (b) Determine the efficiency directly from the results of part (a), and compare with the result
from the previous problem.

9. A 0.05-kg cube of ice at an initial temperature of −15

◦

C is placed in 0.6 kg of water at T = 45

◦

C in

an insulating container of negligible mass. (a) Calculate the final temperature of the water once the
ice has melted. (b) Calculate the change in entropy of the system.

10. (a) Graph a Carnot cycle, plotting Kelvin temperature vertically and entropy horizontally. This is

called a temperature–entropy diagram, or T S-diagram. (b) Show that the area under any curve
representing a reversible path in a temperature–entropy diagram represents the heat absorbed by
the system. (c) Derive from your diagram the expression for the thermal efficiency of a Carnot cycle.

Maciej Wo loszyn

WFiIS AGH

http://fatcat.ftj.agh.edu.pl/~woloszyn/phys/