Chapter 1 - Energy: Facts & Figures

Question 1

Give a definition of energy.

Answer 1

Energy is the quantity of a state of a physical system which describes the capability of the system to perform work.

Question 2

How is work defined mathematically?

Answer 2

W = ∫ F dr, from r_1 to r_2.

Question 3

What is the SI unit of Joule?

Answer 3

J = Nm = kgm^2/s^2

Question 4

What is the SI unit of Watt?

Answer 4

W = J/s = kgm^2/s^3

Question 5

Name some usual units of energy.

Answer 5

Joule (J), 
electronvolt (eV), 
kilowatt hour (kWh), 
calories (cal), 
terawatt year (TWa), 
kilogram coal equivalent (kgCE), 
kilogram oil equivalent (kgOE), 
British thermal unit (BTU)

Question 6

What is the order of magnitude of a terawatt year in terms of kilowatt hours?

Answer 6

1 TWa ≈ 10^13 kWh.

Question 7

What is the value of a kilowatt hour in terms of Joule?

Answer 7

1 kWh = 3.6 * 10^6 J.

Question 8

What is the value of a kilowatt hour in terms of kgCE?

Answer 8

1 kWh = 0.123 kgCE

Question 9

What is the value of an electronvolt in terms of Joule?

Answer 9

1 eV = 1.602 * 10^-19 J

Question 10

How much coal must be burned to provide 1 TWa of energy?

Answer 10

Approximately 10^12 kg.

Question 11

What is the total energy consumption (in 2012)?

Answer 11

2*10^13 kgCE / year.

Question 12

What is the basic metabolic rate of a human being?

Answer 12

80W.

Question 13

What is the definition of a conservative force?

Answer 13

That no work is performed along a closed curve (work performed between two points is path independent).

Question 14

What can be defined for conservative forces, and how is it defined?

Answer 14

A potential energy. It is defined as the negative gradient of the force (- grad F), and the difference in potential energy it two points equals the work being done.

∆E_pot = E_pot(r_2) - E_pot(r_1) = W_12
E_pot = - grad F

Question 15

Derive the expression for the difference in kinetic energy.

Answer 15

The kinetic energy is the work performed to accelerate a mass m.

∆E_kin = ∫ F dr from r_1 to r_2.
= ∫ ma dr = ∫ m * dv/dt * dr = ∫ m * dv * dr/dt = ∫ m * dv * v
= [1/2 mv^2] evaluated from r_1 to r_2
= 1/2 mv_2^2 - 1/2 mv_1^2.

Question 16

What can be said about the total energy in conservative systems?

Answer 16

It is non-dissipative, and the sum of potential and kinetic energy must be constant.

E_tot = E_pot + E_kin.

Question 17

Give examples of dissipative processes.

Answer 17

Friction, turbulence.

Question 18

What happens to classical mechanical energy in dissipative systems? Draw an example of such a process.

Answer 18

Part of it is transformed to undirected inner energy.

Example: a block is dropped to the floor. It starts of with potential energy E_pot = mgh. It gains kinetic energy E_kin = 1/2 mv^2, where E_kin equals the initial potential energy as it hits the ground. At this point each particle will have kinetic energy the same way, which is downward. As it hits the ground the block comes to a rest, but the particles retain the energy in form of random vibrations in the solid.

Question 19

What does the Noether theorem say about energy conservation?

Answer 19

For any symmetry operation that exists, there exists a corresponding conserved quantity.

Question 20

Which conserved quantity arises from translation symmetry?

Answer 20

Momentum.

Question 21

Which conserved quantity arises from rotational symmetry?

Answer 21

Angular momentum.

Question 22

Which conserved quantity arises from translation symmetry in a periodic lattice?

Answer 22

Crystal momentum, wavevector

Question 23

Which conserved quantity arises from time inversion symmetry?

Answer 23

Energy.