05a cpt 5 lecture notes f16 - cehd | umnacademics.cehd.umn.edu/robertson/1-1051/l40-mon-5... ·...

L40-Mon-5-Dec-2016-Sec-5-7-Models-HW39-Moodle-Q31 page 84

L40-Mon-5-Dec-2016-Sec-5-7-Models-HW39-Moodle-Q31


First, find the constant, k, for the iron.

04

4

4

700 80 1000 80

620 920

0.67ln 0.67 4

0.10

kt

k

k

k

u t T u T e

e

ee

kk

Now, find the time to reach 100°F

00.10

0.10

0.10

100 80 1000 80

20 920

0.02ln 0.02 0.10

39

kt

t

t

t

u t T u T e

e

ee

tt

So, the horseshoe will be 100 degrees after 39 minutes.


Iron is different from most other materials (including bronze), in that it does not immediately go from a solid to a liquid at its melting point. For example, water is a solid (ice) at 31°F, and a liquid (water) at 33°F. Iron, by contrast, is a solid at 800°F, but over the next 1,500 °F it becomes increasingly plastic and more "taffy-like" as its temperature increases. This extreme temperature range of variable solidity is the fundamental material property upon which blacksmithing practice depends.

When you borrow money for school or a car or a home you have to pay the person you borrowed the money from for the use of their money. What you pay them is called interest.


Compound Interest is where interest is calculated for a specific period and then that interest is added to the principal when interest is calculated for the next period. It is getting interest on interest.

Let's say there are n compoundings per year.


We keep on going in this manner.


1

ntrA Pn

P = 50 r = 0.06 n = 12 (monthly) t = 68 - 18 = 50 years.

12 50

600

600

0.0650 112

50 1 0.005

50 1.00550 19.94997.00

A

A

AAA

So, is a broken heart and a lost memory worth a thousand dollars in your senior years?

If you are a complete dweeb and this happens every month the savings really pile up:

12 months$1,000 50 years $600,0001 month 1 year


4 3

12

1

0.0875 14

75 1.0275 1.268

59.15

ntrA Pn

P

PP

P

So, you would need $59.15.


0.06 50

350

5050 20.091004.50

rtA PeA eA eAA

With monthly compounding the final amount was $997. So, with continuous compounding you earn an additional $7.50.

0.06

0.06

0.06

3

3

ln3 ln

ln3 0.061.0990.0618.3

rt

t

t

t

A PeP Pe

e

e

t

t

t


So, the first option is the better one.


Logarithmic Scale A logarithmic scale is a scale of measurement that displays the value of a physical quantity using intervals corresponding to orders of magnitude (powers of 10), rather than a standard linear scale. Scientists often use logarithms to rescale measurements of objects or phenomena when the measurements are very small or very large. For example, the pH of Shampoo is about

107.4 10 molesper liter while a moderate earthquake will release energy on the order of

224 10 Newton-meters . Numbers of these sizes are difficult to get a “feel” for using a linear scale.

Earthquakes An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The slip between two tectonic plates along a transform fault, such as the San Andres fault in California, is a common cause of such a release of energy where one plate moves relative to the other (the Pacific Plate is moving north and being pushed under the North American Plate at the Aleutian Islands near Alaska). There are two important types of seismic waves: the p-wave and the s-wave. The primary seismic wave, or p-wave, is like a sound wave traveling through the earth. It is a longitudinal pressure wave that moves the ground back and forth in the direction of propagation. The secondary seismic wave, or s-wave, is like ripple on a pond. It is a transverse wave that moves the ground perpendicular to the direction of the wave propagation. Earthquakes also produce a surface wave that travels along the surface of the earth, rather than through it like the p and s waves.

The Richter magnitude scale was developed to assign a single number to quantify the energy that is released during an earthquake. The higher the number the more energy released and the higher the potential for ground shaking and hence damage to life and property.

The scale is a base-10 logarithmic scale. The Richter magnitude is defined as the logarithm of the ratio of the amplitude A of seismic waves measured by a seismograph to an arbitrary small amplitude, 0A

0logR

A

A

An earthquake that measures 5.0 on the Richter scale has a shaking amplitude 10 times larger than one that measures 4.0, and corresponds to a 32 times larger release of energy.

The Richter scale was developed at the California Institute of Technology in 1935 by Charles Richter and Beno Gutenberg. Richter arbitrarily chose a magnitude 0 event to be an earthquake that would show a maximum combined horizontal displacement of 1 µm (0.00004 inch) on a seismogram recorded using a Wood-Anderson torsion seismograph 62 miles from the earthquake epicenter. This choice was intended to prevent negative magnitudes from being assigned. The smallest earthquakes that could be recorded and located at the time were around magnitude 3. However, the Richter scale has no lower limit, and sensitive modern seismographs now routinely record quakes with negative magnitudes.

The following chart will give you an idea of the energy released by an earthquake and the damage it can cause.


Magnitude Type Average earthquake effects and examples Estimated

frequency per year

Less than 2.0

Micro Microearthquakes are not felt by people. Continual

Several million

2.0–2.9

Minor

Felt slightly by some people. No damage to buildings.

One million

3.0–3.9

Often felt by people, but rarely cause damage. Shaking of indoor objects is noticeable. Oklahoma City bombing, 1995 (equivalent to ½ ton of TNT)

100,000

4.0–4.9 Light

Felt by most people. Noticeable shaking of indoor objects and rattling noises. Generally causes none to minimal damage. Explosion at Chernobyl nuclear power plant, 1986 (equivalent to 10 tons of TNT)

10,000 to 15,000

5.0–5.9 Moderate Felt by everyone. Can cause damage to poorly constructed buildings with none to slight damage to other buildings.

1,000 to 1,500

6.0–6.9 Strong

Felt in wider areas up to hundreds of miles from the epicenter. Damage to some well-built structures. Earthquake-resistant structures survive with slight damage. Violent shaking in epicentral area. Death toll up to 25,000. Atomic Bomb dropped on Hiroshima, 1945 (equivalent to 15,000 tons of TNT)

100 to 150 per year

7.0–7.9 Major

Felt across great distances with major damage mostly limited to 100 miles from epicenter. Damage to most buildings. Death toll up to 250,000.

10 to 20 per year

8.0–8.9

Great

Felt in extremely large regions. Major damage to buildings, structures likely to be destroyed. Death toll ranges from 1,000 to 1 million. San Francisco earthquake, 1906 (equivalent to 15 million tons of TNT) and Krakatoa 1883 (equivalent to 200 million tons of TNT)

One per year

9.0–9.9

Severe damage or collapse of all buildings. Heavy damage and shaking extends to distant locations. Permanent changes in ground topography. Death toll usually over 50,000. Anchorage earthquake, 1964 (equivalent to 950 million tons of TNT)

One per 10 to 50 years

10.0 Humongous Never recorded

11.0 OMG Never recorded


12.6 Kiss your sorry ass goodbye

Utter devastation. Never recorded but evidence shows one happened 65 million years ago when a meteor hit the Yucatán Peninsula creating the Chicxulub crater (equivalent to 100 trillion tons of TNT). Mass extinction of life all over the earth.

Hopefully, only in the past

The 1906 San Francisco earthquake had an amplitude of 82 10 times the reference amplitude, 0A .

In 1989 the World Series earthquake in San Francisco had an amplitude of 68 10 times the reference amplitude, 0A . Here are the Richter numbers:

0

0 0

88

1906

2 10log log log 2 log 10 0.3 8 8.3AA AAR

0

0 0

66

19898 10

log log log 8 log 10 0.9 6 6.9A

A A

AR

The ratio of the amplitudes of the two earthquakes is

821906

61989

2 100.25 10 25

8 10

A

A. This means

the 1906 earthquake released 25 times the energy of the 1989 earthquake even though the Richter numbers differ by only 8.3 6.9 1.4 .

Sound The loudness of sound can be measured using a decibel scale, named after Alexander Graham Bell. Loudness, L , is defined in a way that is similar to the Richter scale:

010 log

I

IL

where I is the intensity of the sound in W atts

square meter and

12

010 Watts

square meterI , which is the intensity

of the softest sound that can be heard by humans. This formula enables us to assign a numeric value to a subjective attribute of sound, which we call “loudness.” A Watt is a unit of power so Watts per square meter measures the energy transfer from the sound wave to the surface of the ear drum. This causes the ear drum to vibrate, sending electrical signals to the brain, which it interprets as sound. Too little energy transfer and the signals are not generated; too much energy transfer will cause damage to the eardrum, which results in pain.

Here are three examples:

1. The intensity of sound for normal conversation is 1 millionth of a W att

square meter. The decibel level is

66 6

121 10

10 log 10 log 10 log 1 10 10 log 10 10 6 601 10o

IL dbl

I


2. The intensity of sound at a rock concert can reach half a W att

square meter. The decibel level is

11

11

25 10

10 log 10 log 5 1010

10 log 5 11 10 0.7 11 117 decibelsL

3. The threshold of pain in a human is 1 W att

square meter, The decibel level is

12

12110 log 10 log 10 10 12 120 deci

10belsL

So, if the concert is difficult to listen to it may be the loudness rather than the ineptitude of the bands.

Acidity The pH of a compound is the measure of the hydrogen ion concentration. A high pH indicates an acid while a low pH indicates a base. Concentrated hydrochloric acid has a pH of 0, which indicates

extreme acidity. Pure water has a pH of 7, which is neutral. Concentrated sodium hydroxide, a base, has a pH of 14, which

indicates extreme alkalinity.

pH is defined mathematically as the common logarithm of the reciprocal

of the hydrogen ion activity. That is,

+H

H

1pH log log a

a. The

hydrogen ion activity is measured by its concentration using units of moles per liter, which is called its molarity. A mole measures the amount of a substance that is present. For example, in one mole of hydrogen

there are 236.02 10 hydrogen atoms. 236.02 10 is called Avogadro’s number in honor of the 19th century Italian scientist Amedeo Avogadro.

In pH, the p denotes a negative logarithm and the H denotes hydrogen. Here is a chart of pH values for various substances.

As an example, let’s find the pH of a 2 liter solution that contains 5 grams of nitric acid 3HNO ?

First, calculate the molarity (moles per liter) of the solution. To do this, convert the 5 grams of 3HNO

into moles by dividing by its molar mass:

5 gm

0.079 mole63 gm/mole

Next, divide by the volume: 0.079 mole

0.040 mole/liter2 liter

The pH of the solution is the negative logarithm of the molarity.

pH log 0.040 1.40 1.40

05a cpt 5 lecture notes f16 - cehd | umnacademics.cehd.umn.edu/robertson/1-1051/l40-mon-5... ·...

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