Unit 15. Kinetic energy. Power and Efficiency.

Learn the following words and word combinations by heart:

capacity	нагрузка; функциональные возможности, ёмкостное сопротивление
convert (into)	превращать
dissipate (to)	рассеиваться (об энергии или мощности), исчезать
efficiency electrical ~ mechanical ~ overall ~ thermal ~	коэффициент полезного действия, КПД; отношение произведённой работы к использованной энергии полный кпд (сети); электрический коэффициент полезного действия механический кпд суммарный коэффициент полезного действия всей установки; полная производительность, термический  кпд; КПД  теплового  двигателя
energy ~ output accumulation of ~ deliver ~ depletion of ~ electrical ~ loss large amount of ~ loss of ~ thermal ~ loss	потреблённая электроэнергия; электроэнергия выходная мощность, вырабатываемая  энергия; выдача энергии накапливание энергии поставлять, передавать энергию уменьшение, расход энергии потери электроэнергии, энергетические  потери большое количество энергии потери энергии потери тепловой энергии (или термической энергии)
kinetic friction	трение движения, кинетическое трение
operate uniformly	функционировать непрерывно, постоянно
point of application of the force	точка приложения силы
power develop a ~	питание, электроснабжение; мощность; энергия достигать, развивать мощность
surroundings	естественные условия
total work	совокупная, суммарная, общая работа
unit	единица физической величины
unity	единица (число), единое целое

Ex.1. Look at Appendix 1 and read the following mathematical symbols and abbreviations.

T = ½ m v² ; U _1-2 = T₂-T₁ = ∆ T; T₁ + U _1-2 = T₂; e_{m =} P _{output :} P _input

Part 1. Kinetic energy.

The kinetic energy T of the particle is defined as

T = ½ m v² (3)

and is the total work which must be done on the particle to bring it from a state of rest to a velocity v. Kinetic energy T is a scalar quantity with the units of N•m or joules (J). Kinetic energy is always positive, regardless of the direction of the velocity. Equation 3 may be restated as

U _1-2 = T₂-T₁ = ∆ T. (4)

which is the work-energy equation for a particle. The equation states that the total work done by all forces acting on a particle during an interval of its motion from condition 1 to condition 2 equals the corresponding change in kinetic energy of the particle. Although T is always positive, the change ∆ T may be positive, negative, or zero.

When written in this concise form, Eq. 4 tells us that the work always results in a change of kinetic energy.

Alternatively, the work-energy relation may be expressed as the initial kinetic energy T₁ plus the work done U _1-2 equals the final kinetic energy T₂ or

T₁ + U _1-2 = T₂ (4 a)

When written in this form, the terms correspond to the natural sequence of events. Clearly, the two forms 4 and 4 a are equivalent.

We now see from Eq. 4 that a major advantage of the method of work and energy is that it avoids the necessity of computing the acceleration and leads directly to the velocity changes as functions of the forces which do work. Further, the work-energy equation involves only those forces which do work and thus give rise to changes in the magnitude of the velocities.

We consider now two particles joined together by a connection which is frictionless and incapable of any deformation. The forces in the connection constitute a pair of equal and opposite forces, and the points of application of these forces necessarily have identical displacement components in the direction of the forces. Hence, the net work done by these internal forces is zero during any movement of the system of the two connected particles. Thus, Eq. 4 is ap­plicable to the entire system, where U _1-2 is the total or net work done on the system by forces external to it and ∆T is the change, T₂-T₁ in the total kinetic energy of the system. The total kinetic energy is the sum of the kinetic energies of both elements of the system. It may now be observed that a further advantage of the work – energy method is that it permits the analysis of a system of particles joined in the manner described without dismembering the system.

Application of the work-energy method calls for an isolation of the particle or system under consideration. For a single particle a free-body diagram showing all externally applied forces should be drawn. For a system of particles rigidly connected without springs, an active – force diagram that shows only those external forces which do work (active forces) on the entire system may be drawn.

Comprehension check. (Part 1.)

Ex. 1. Answer the following questions.

1. How is the kinetic energy T of the particle defined?

2. What does the work-energy equation for a particle state?

3. What does the work always result in?

4. What is a major advantage of the method of work and energy?

5. Which forces does the work-energy equation involve?

6. What does the application of the work-energy method call for?

7. What diagram should be drawn for a single particle?

8. What diagram should be drawn for a system of particles?

Ex.2. Say whether the following statements are True or False.

1. Kinetic energy T is a vector quantity with the units of N•m or joules (J).

2. Kinetic energy is always positive, regardless of the direction of the velocity.

3. As T is always positive, the change ∆ T must be positive as well.

4. The work-energy equation involves all the forces acting on the body and thus give rise to changes in the magnitude of the velocities.

5. The total kinetic energy is the sum of the kinetic energies of both elements of the system.

6. For a single particle an active – force diagram that shows only forces which do work should be drawn.

Ex. 3. Insert the verbs in an appropriate form into the gaps.