Consider a traveling wave described by the formula

$y_1(x,t) = A \sin(k x - \omega t)$ .

This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

Part A

Which one of the following statements about the wave described in the problem introduction is correct?

ANSWER:

The wave is traveling in the

direction.
The wave is traveling in the

direction.
The wave is oscillating but not traveling.
The wave is traveling but not oscillating.

Part B

Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time t=0

this new wave should have the same displacement as y_1(x,t)

, the wave described in the problem introduction.

ANSWER:

$A \cos (k x - \omega t)$
$A \cos (k x + \omega t)$
$A \sin (k x - \omega t)$
$A \sin (k x + \omega t)$

The principle of superposition states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves , where is the wave described in Part A and is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

$y_{\rm s}(x,t) = y_{\rm e}(x) y_{\rm t}(t)$ .

This form is significant because , called the envelope, depends only on position, and depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of .

Part C
Find and . Keep in mind that should be a trigonometric function of unit amplitude.

erinhyena808

Materials fracture when the force per unit area, called STRESS,exceeds a critical value. In general, the critical stress dependson the direction as well as the magnitude of the force. Forexample, bones break differently when subjected to torsional(twisting) stress than when subjected to compressional (orsqueezing) stress. Let us consider compressional stress. Wet humanbone for 20-39 year olds has an ultimate compressive strength of ~16x104Ncm2 [8]. Consider a person who falls from a height to theground. Upon landing, the person's kinetic energy is converted towork. If the person decelerates over a distance h, we can estimatethe average force during the collision by

If a person lands stiff-legged on a hard surface and doesn'tbounce, the deceleration occurs over a very short distance, h ~ 1cm. Since the force is transmitted up the leg, the stress isgreatest where the cross-sectional area is least, i.e., the tibiajust above the ankle [8].
The following lists some different conditions that may occur duringlanding and physical explanations for why they make bone fracturefrom a fall more or less likely [8]:

(a) bounce [more likely. The impulse is twice as large for anelastic collision as for an inelastic collision.]
(b) bent knees [less likely. Bending the knees increases the timeand distance over which deceleration occurs, which reduces theforce.]
(c) landing in loose earth or sand [less likely. This increases hwhich reduces F.]
(d) turning the body to land on the side [less likely. Thisincreases h. It also distributes the force widely over the body(increases A), thereby reducing the stress.]

So, why can a properly executed karate kick break a concrete blockwithout fracturing bones [16]? First, bone is a very strongmaterial. Its ultimate compressive strength is approximately 40times larger than concrete. Second, contact is made with the edgeof the foot. This concentrates the force into a small area of thetarget and reduces the likelihood of bending a bone to the point offracture. Third, the collision with the target is essentiallyinelastic and extends over several milliseconds, so the peak force,though large, does not exceed the strength of the bone. If A(cross-sectional area of the tibia) ~ 3.3 cm2, compute how far a 69kg person can fall and land stiff-legged on both legs withoutbreaking a bone. Assume F is split evenly between two legs. (Hint:F<or =2(theta)A )

Vapor cycles
5EM - Thermodynamics
September 2010
1 The T-S diagram
The temperature-entropy diagram is an alternative to the well knownpressure-
volume diagram. Before using it to analyze different vapor cycles,we need
to understand it properly.
1. Plot the following transformations for ideal gases in a T-Sdiagram
(a) Isothermal expansion
(b) Isobaric expansion
(c) Isometric heating
(d) Reversible adiabatic expansion
(e) Irreversible adiabatic expansion
2. Repeat the last question, but this time do it for a realsubstance,
starting in the compressed liquid1 region and ending in thesuperheated
region, when possible. Indicate the boundaries of eachregion.
3. We know that under certain circumstances the area under theplot
in a P-V diagram represents the work performed during theprocess.
What does it represent in this case? Under what assumptions isyour
answer true? Compare examples 6.1 and 6.2.
2 Thermal machines
We understand by thermal machine a device capable of transformingther-
mal energy (heat) into mechanical energy (work). We already knowthat
these machines have a limited efficiency which, in the two-sourcesmodel, is
obtained by Carnotâs cycle.
(Compressed liquid is sometimes also called subcooled liquid)
2.1 Carnot cycle
This cycle, although efficient, is impractical. To begin with, theextreme
temperatures of the cycle should be within the sources temperaturesfor a
finite heat flow to exist.
1. Given the temperature of the sources, draw the cycle accordingto the
named limitation.
2. Show that this leads to lower efficiency than the theoreticallimit.
3. Assume that heat flow from/to the sources is proportional to thetem-
perature difference between them and the working fluid, anddemon-
strate that a higher power output is related to a decreasedefficiency.
2.2 Rankine cycle
Out of the several practical possibilities, Rankine cycle and itsvariations is
widely used, especially for electric power generation.
1. Draw the basic cycle in a T-S diagram and explain it.
2. What does the efficiency of the cycle depend on? Comparewith
Carnotâs.
3. What are the differences between the ideal cycle and the realone?
4. Describe how it is modified by superheating and/or reheating.Explain
te benefits.
2.3 Problems
After answering the previous questions solve the followingexercises from the
book: 8.2, 8.4, 8.5, 8.13, 8.19, 8.20
3 Frigorific machines
3.1 Reverse Carnot cycle
We have already studied Carnot cycle in detail as the ideal thermalmachine.
Now we will take is as the starting point to understand thecharacteristics
of frigorific machines.
1. Draw the ideal cycle in a T-S diagram and explain it.
2. Show that this cycle has a theoretical efficiency Ã =Tcold/Thot-Tcold
3. What are the difficulties in implementing such cycle in actualdevices?
3.2 Actual refrigerators
1. Why is the turbine replaced by an expansion valve in mostcases?
2. Plot the actual cycle and explain the differences with the idealcase.
3. How do cascade and multistage systems improve the performanceof
the refrigerator?
4. What aspects are important in the choice of a refrigerant fluid?Justify
3.3 Problems
After answering the previous questions solve the followingexercises from the
book: 10.2 10.7, 10.8, 10.9, 10.22, 10.23 10.1D

PHL 101 Lecture : BookI&BookII

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