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Modern Engineering Thermodynamics

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Problems 55<br />

and<br />

Death rate = α 2 + β 2 N<br />

Ignoring the effects of immigration into the population,<br />

determine the rate balance equation for this population and<br />

show that it can be solved to produce the following population<br />

function:<br />

α/β<br />

N =<br />

ð1 + e −αt Þ<br />

where α = α 1 – α 2 and β = β 1 + β 2 . Note that this model predicts a<br />

limiting population size of α/β as t → ∞.<br />

44. Is the amount of gold reserves held by a nation a conserved<br />

quantity? Explain what happens if more currency is put into<br />

circulation while the currency base (e.g., gold reserves) is held<br />

constant.<br />

45. Use the balance concept to explain the changes in the wealth of<br />

a nation. In particular, describe methods by which a nation can<br />

add or lose wealth by transport across its boundaries, and show<br />

how it can produce or destroy wealth within its boundaries.<br />

46. Are the natural resources of a nation conserved in a<br />

thermodynamic sense? If not, explain what would have to be<br />

done to cause them to be conserved. Give a specific example<br />

where this is currently being done.<br />

47. Using Eq. (2.9), write a balance equation for the total potential<br />

energy of a system during a time interval δt. Is this potential<br />

energy conserved? How can potential energy be produced or<br />

destroyed within a system?<br />

48. From Eq. (2.9), develop a balance equation for the gain in<br />

kinetic energy of a system during a time interval δt. Is kinetic<br />

energy a conserved quantity? How can kinetic energy be<br />

produced or destroyed within a system?<br />

49. Create a balance equation for the change in the number<br />

of chairs in a classroom during a time interval δt using<br />

Eq. (2.9). Are classroom chairs a conserved quantity? Describe<br />

how classroom chairs can be transported into and out of the<br />

classroom and produced or destroyed within the classroom.<br />

50. Using Eq. (2.9), prepare a balance equation for the gain in the<br />

number of dollar bills in your pocket during a week. Are these<br />

dollar bills a conserved quantity? How can dollar bills be<br />

transported, created, or destroyed in your pocket?<br />

51. Equation (2.10) provides a net balance equation for any quantity.<br />

Use this equation to construct a net balance equation for the<br />

change in the number of automobiles contained within the city<br />

limits of Detroit, Michigan, during a time interval of 1 year. What<br />

are some possible mechanisms for transport, production, and<br />

destruction of automobiles?<br />

52. Reproduction and death are production and destruction<br />

mechanisms for humans. Using Eq. (2.10), develop a net<br />

balance equation to predict the net gain in people in any given<br />

family group over a time interval of 10 years.<br />

53. Take a typical library as your system. Using Eq. (2.10), develop<br />

a net balance equation for the net gain in books in the library<br />

over the period of 6 months. Specify explicit transport,<br />

production, and destruction mechanisms for the books.<br />

54.* Determine the muzzle velocity of a 10.0 g bullet that impacts a<br />

5.00 kg ballistic pendulum 0.200 m below the fulcrum of the<br />

pendulum and produces a 25.0° deflection.<br />

55.* You are designing a ballistic pendulum for a baseball throwing<br />

contest. The baseballs have a mass of 0.142 kg and the<br />

pendulum is a 10.0 kg horizontal hollow cylinder closed on the<br />

far end and suspended from the ceiling. Determine the length of<br />

the suspension cords if the deflection of the cylinder is not to<br />

exceed 20.0° when the baseball impact velocity is 40.0 m/s<br />

(89.5 mph).<br />

56. The muzzle velocity of a Daisy Red Ryder BB gun is 260 ft/s and<br />

the mass of a BB is 7.5 × 10 −4 lbm. Determine the angle of<br />

deflection of a 0.50 lbm ballistic pendulum suspended 2.0 ft<br />

from its support when impacted by the BB.<br />

57. In 1945, the United States Army developed the largest artillery<br />

weapon ever constructed. Called the Little David (Figure 2.14),<br />

it has a bore of 900. mm and fires a 2.00 ton shell with a<br />

muzzle velocity of 200. ft/s that produces a crater 38.0 ft wide<br />

and 20.0 ft deep. As chief engineer of Army Ordnance, you are<br />

to design a ballistic pendulum for this gun. If the distance from<br />

the fulcrum of the pendulum to the point of impact is 10.0 ft,<br />

how much mass would the pendulum have to contain if<br />

deflection of its suspension cord is not to exceed 20.0°?<br />

FIGURE 2.14<br />

Problem 57.<br />

58.* The Sandia National Laboratory hypervelocity two-stage light gas<br />

gun produces muzzle velocities of 12 km/s with 5.0 g<br />

projectiles. A ballistic pendulum is to be designed for this gun.<br />

It is to be suspended on cords 1.0 m long to produce a 15°<br />

deflection when impacted by the gun’s projectile. Determine the<br />

mass required for the pendulum.<br />

59. Since mass is a conserved quantity, use Eq. (2.21) to develop a<br />

conservation of mass balance equation over a time interval of<br />

1 year for a system consisting of the entire Earth (do not forget<br />

to draw a sketch of your system).<br />

60. A person is trying to fill a 10.0 gallon bucket by hand with a<br />

dipper (Figure 2.15). One dipper of water is added each second,<br />

and the dipper holds 1.0 lbm of water. Unfortunately, the bucket<br />

has a hole in it and water leaks out at a rate of 0.50 lbm/s. How<br />

long does it take for the person to fill the bucket? (Note: Water<br />

weighs 8.3 lbf/gal at standard gravity.)

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