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LSP 121 -- 10/12/10

Midterm Exam

• Discuss Midterm Exam

Proportional Units

• Proportional units are units that can be converted from one to another by multiplying by a constant conversion factor.

• Example: To convert from feet to yards, use the conversion factor 0.33333 yards / foot.

• For Homework 3, all conversions of linear units should be done with units to show that you are obtaining the correct answer. To convert 24 feet to yards:

  24 feet × (1 / 3) yard / foot = (24 × 1 / 3)(foot × yard / foot) = 8 yards

• Metric Prefixes: Commonly Used Prefixes   Complete List

• Until 1959, the U.S. meter was officially defined as 39.34 inches.

• Beginning in 1959, the international inch was defined as 2.54 centimeters.

• 1 pound = 453.59237 grams = 0.45359237 kilograms

• 1 U.S. gallon = 3.7854118 liters.

• English Units of Length
  1 foot = 12 inches    1 yard = 3 feet    1 mile = 1760 yards = 5280 feet

• English Units of Weight
  1 pound = 16 ounces    1 ton = 2,000 pounds

• Units of Time
  1 minute = 60 seconds    1 hour = 60 minutes    1 day = 24 hours

  1 year = 365 days, 6 hours, 9 minutes, 9.54 seconds

Example Calculations

Here is an Excel file showing the calculations: proportional-units.xls, 4-26 Review Questions.

1. Convert one million seconds to days.

   Ans: 10^6 sec × 1/(60 sec/min) × 1/(60 min/hour) × 1/(24 day/hour) = 11.57 days.

2. Convert 75 miles per hour to meters per second.

   Ans:
   75 mi/h × (1/3600) h/s × 5280 ft/mi × 12 in/ft × 2.54 cm/in × (1/1)m/cm =
      75 × (1/3600) × 5280 × 12 × 2.54 × (1/100) ×
      (mi/h) × (h/s) × (ft/mi) × (in/ft) × (cm/in) × (m/cm) = 33.53 m/sec

3. Convert 1,000 Euro to Japanese Yen. Use the current rates using the exchange rates 0.737 Euro / Dollar and 94.03 Yen / Dollar.

   Ans:
   1000 euro × (1/0.773187) dollar / euro × 121.468 yen / dollar =
       1000 × (1/0.773187) × 121.468 × euro × (dollar/euro) × (yen/dollar) = 157,100.42 yen.

4. How many miles are in a light year. First convert years to seconds. Then use the official definition to determine how far light travels in one second and multiply. Assume that there are exactly 356.25 days in a year.

   Ans:
   299,792,458 m/sec * 100 cm/m * (1/2.54) in/cm * (1/12) ft/in * (1/5280) mile/ft *
   60 sec/min * 60 min/hr * 24 hr/day * 365.25 day/yr =
   5.87863 × 10¹² miles/year

5. Compute the circumference of the earth in inches using the 1793 definition of the meter.

   Ans:
   40,000,000 m × (1 in / 2.54 cm) × 100 cm / m =
       40,000,000 × (1 / 2.54) × 100 × m × (in / cm) × (cm / m) = 1.575 × 10¹⁰ in

Homework 3

• Look at Homework 3a and Homework 3b.

Logarithms

• Recall that the logarithm of a positive number x is the value s that solves the equation x = 10^s.

• We write s = log x.

• Examples:
  log 1000 = log 10³ = 3
  log 100,000,000 = log 10⁸ = 8
  log 10 = log 10¹ = 1
  log 1 = log 10⁰ = 0
  log 429 = log 10^2.632467 = 2.632467

• Use Excel to compute log 429:
  Enter 429 into cell A1.

  Enter =LOG(A1) into cell B1.

  The logarithm 2.632467 appears in cell B1.

Antilogarithms

• If you know the logarithm s and want to determine the original number x, just compute x = 10^s.

• Examples. Given the logarithm, determine the original number:
  Logarithm: 5, Original Number: 10⁵ = 100,000.

  Logarithm: 2, Original Number: 10² = 100.

  Logarithm: 0, Original Number: 10⁰ = 1.

  Logarithm: 0.5, Original Number: 10^0.5 =
      10^1/2 = Square Root of 10 = 3.162278.

• Use Excel to compute the antilog of 3.89:
  Enter 3.89 cell C1.

  Enter =10^C1 into cell D1.

  The antilogarithm 7762.471 appears in cell D1.

Logarithmic Units

• Logarithmic units are often used when the ratio of the largest to the smallest values of the phenomenon being studied is many orders of magnitude. Examples: sound heard by the human ear, brightnesses of stars, intensities of earthquakes, strength of acids. Perhaps shows the versatility the human senses of hearing, sight, feeling and taste.

• The Richter Scale measuring the strength of earthquakes is a logarithmic scale.

• An earthquake that measures 4.5 on the Richter Scale is 10 times as powerful as a 3.5 earthquake.

• To determine the Richter Scale R from the intensity I, compute the logarithm R = log I.

• To determine the intensity I from the Richter Scale R, compute the antilogarithm: I = 10^R.

• Here are some other units that have logarithmic scale:
  Decibels for measuring sound intensity: A sound of 30 decibels is 10 times louder than a sound of 30 decibels.

  Magnitudes of celestial objects: a star of magnitude 4 is 100^1/5 = 2.512 times fainter than a star of magnitude 3.

  A magnitude zero star is approximately the brightness of Polaris, the North Star. Here is a celestial map of the region containing the North Star. The human eye cannot usually see objects of magnitude greater than 5 or 6.

  Here are the magnitudes of some of the brightest objects in the sky:

  Sirius	-1.42
  Saturn	0.7
  Jupiter	-2.8
  Mars	-2.9
  Venus	-4.7
  Full Moon	-12.6
  Sun	-26.7

  pH value for acids or bases: a liquid that has a pH of 9 is 10 times more acidic that a liquid with a pH of 8.0.

Practice Problems

1. If the intensity of an earthquake is 781,000, what is its Richter number?

   Ans: log(781,000) = 5.89

2. If the Richter number of an earthquake is 6.2, what is its intensity?

   Ans: 10^6.2 = 1,585,000

3. If the intensity of a sound is 5,342,000 times the threshold of hearing, what does it measure in decibels?

   Ans: 10 * log(5,342,000) = 67.3

4. The Krakatoa volcanic eruption of 1883 is probably the loudest sound that occurred on earth in recent history. Scientists estimate that it measured 310 decibels. How many times greater is its intensity than the threshold of hearing?

   Ans: 10^{310 / 10} = 10³¹

5. The sound of someone talking twenty miles away is 1,000 times less intense than the threshold of hearing. What does this sound measure in decibels.

   Ans: 10 * log(1/1000) = 10 * (-3) = -30