Modern Engineering Thermodynamics

Problems 55
and
Death rate = α 2 + β 2 N
Ignoring the effects of immigration into the population,
determine the rate balance equation for this population and
show that it can be solved to produce the following population
function:
α/β
N =
ð1 + e −αt Þ
where α = α 1 – α 2 and β = β 1 + β 2 . Note that this model predicts a
limiting population size of α/β as t → ∞.
44. Is the amount of gold reserves held by a nation a conserved
quantity? Explain what happens if more currency is put into
circulation while the currency base (e.g., gold reserves) is held
constant.
45. Use the balance concept to explain the changes in the wealth of
a nation. In particular, describe methods by which a nation can
add or lose wealth by transport across its boundaries, and show
how it can produce or destroy wealth within its boundaries.
46. Are the natural resources of a nation conserved in a
thermodynamic sense? If not, explain what would have to be
done to cause them to be conserved. Give a specific example
where this is currently being done.
47. Using Eq. (2.9), write a balance equation for the total potential
energy of a system during a time interval δt. Is this potential
energy conserved? How can potential energy be produced or
destroyed within a system?
48. From Eq. (2.9), develop a balance equation for the gain in
kinetic energy of a system during a time interval δt. Is kinetic
energy a conserved quantity? How can kinetic energy be
produced or destroyed within a system?
49. Create a balance equation for the change in the number
of chairs in a classroom during a time interval δt using
Eq. (2.9). Are classroom chairs a conserved quantity? Describe
how classroom chairs can be transported into and out of the
classroom and produced or destroyed within the classroom.
50. Using Eq. (2.9), prepare a balance equation for the gain in the
number of dollar bills in your pocket during a week. Are these
dollar bills a conserved quantity? How can dollar bills be
transported, created, or destroyed in your pocket?
51. Equation (2.10) provides a net balance equation for any quantity.
Use this equation to construct a net balance equation for the
change in the number of automobiles contained within the city
limits of Detroit, Michigan, during a time interval of 1 year. What
are some possible mechanisms for transport, production, and
destruction of automobiles?
52. Reproduction and death are production and destruction
mechanisms for humans. Using Eq. (2.10), develop a net
balance equation to predict the net gain in people in any given
family group over a time interval of 10 years.
53. Take a typical library as your system. Using Eq. (2.10), develop
a net balance equation for the net gain in books in the library
over the period of 6 months. Specify explicit transport,
production, and destruction mechanisms for the books.
54.* Determine the muzzle velocity of a 10.0 g bullet that impacts a
5.00 kg ballistic pendulum 0.200 m below the fulcrum of the
pendulum and produces a 25.0° deflection.
55.* You are designing a ballistic pendulum for a baseball throwing
contest. The baseballs have a mass of 0.142 kg and the
pendulum is a 10.0 kg horizontal hollow cylinder closed on the
far end and suspended from the ceiling. Determine the length of
the suspension cords if the deflection of the cylinder is not to
exceed 20.0° when the baseball impact velocity is 40.0 m/s
(89.5 mph).
56. The muzzle velocity of a Daisy Red Ryder BB gun is 260 ft/s and
the mass of a BB is 7.5 × 10 −4 lbm. Determine the angle of
deflection of a 0.50 lbm ballistic pendulum suspended 2.0 ft
from its support when impacted by the BB.
57. In 1945, the United States Army developed the largest artillery
weapon ever constructed. Called the Little David (Figure 2.14),
it has a bore of 900. mm and fires a 2.00 ton shell with a
muzzle velocity of 200. ft/s that produces a crater 38.0 ft wide
and 20.0 ft deep. As chief engineer of Army Ordnance, you are
to design a ballistic pendulum for this gun. If the distance from
the fulcrum of the pendulum to the point of impact is 10.0 ft,
how much mass would the pendulum have to contain if
deflection of its suspension cord is not to exceed 20.0°?
FIGURE 2.14
Problem 57.
58.* The Sandia National Laboratory hypervelocity two-stage light gas
gun produces muzzle velocities of 12 km/s with 5.0 g
projectiles. A ballistic pendulum is to be designed for this gun.
It is to be suspended on cords 1.0 m long to produce a 15°
deflection when impacted by the gun’s projectile. Determine the
mass required for the pendulum.
59. Since mass is a conserved quantity, use Eq. (2.21) to develop a
conservation of mass balance equation over a time interval of
1 year for a system consisting of the entire Earth (do not forget
to draw a sketch of your system).
60. A person is trying to fill a 10.0 gallon bucket by hand with a
dipper (Figure 2.15). One dipper of water is added each second,
and the dipper holds 1.0 lbm of water. Unfortunately, the bucket
has a hole in it and water leaks out at a rate of 0.50 lbm/s. How
long does it take for the person to fill the bucket? (Note: Water
weighs 8.3 lbf/gal at standard gravity.)