By what transfer mechanisms does energy enter and leave (a) your television set? (b) Your gasoline-powered lawn mower? (c) Your hand-cranked pencil sharpener?
By what transfer mechanisms does energy enter and leave (a) your television set? (b) Your gasoline-powered lawn mower? (c) Your hand-cranked pencil sharpener?
By what transfer mechanisms does energy enter and leave (a) your television set? (b) Your gasoline-powered lawn mower? (c) Your hand-cranked pencil sharpener?
(a)
Expert Solution
To determine
The energy transfer mechanism by which energy enter and leave the television set.
Answer to Problem 8.1QQ
The energy enters in the form of electrical energy, electromagnetic radiation and energy leaves in the form of heat, mechanical waves and electromagnetic radiation.
Explanation of Solution
In the television set, the energy enters in the form of electrical energy for the power and in the form of the electromagnetic radiation for the signal purpose.
The energy leaves by the television in different forms. First type of the energy is left in the form of the heat from the hot surfaces of the television into the air, second type of the energy is left in the form of mechanical waves as sound from the speaker and third type of energy form is left in the form of electromagnetic radiation from the television screen.
Conclusion:
Therefore, the energy enters in the form of electrical energy, electromagnetic radiation and energy leaves in the form of heat, mechanical waves and electromagnetic radiation.
(b)
Expert Solution
To determine
The energy transfer mechanism by which energy enter and leave the gasoline-powered lawn mower.
Answer to Problem 8.1QQ
The energy enters in the form of fuel energy and energy leaves in the form of heat, mechanical waves and work.
Explanation of Solution
In the gasoline-powered lawn mower, the energy enters in the form of fuel energy by using gasoline for the power.
There are three main ways in which the energy leaves the gasoline-powered lawn mower. The energy leaves in the form of the heat from the hot surfaces into air, in the form of mechanical waves as sound and in the form of work on the blades of the grass cutter.
Conclusion:
Therefore, the energy enters in the form of fuel energy and energy leaves in the form of heat, mechanical waves and work.
(c)
Expert Solution
To determine
The energy transfer mechanism by which energy enter and leave the hand-cranked pencil sharpener.
Answer to Problem 8.1QQ
The energy enters in the form of work in turning the crank and energy leaves in the form of work done on the pencil and mechanical waves as sound.
Explanation of Solution
In the hand-cranked pencil sharpener, the energy enters in the form of the form of work in turning the crank by hand.
There are two main ways in which the energy leaves the hand-cranked pencil sharpener. The energy leaves in the form of mechanical waves as sound from and in the form of work done on the pencil.
Conclusion:
Therefore, the energy enters in the form of work in turning the crank and energy leaves in the form of work done on the pencil and mechanical waves as sound.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
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