TECH1005 Physical Units

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TECH1005Physical Units

• Jon Ivins
• DMU

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Trivial but important to understand

• Suppose I travel 8000 ft in 1 minute. What is my
  speed in mph?
• Easy to say speeddistance / time
• So speed 8000 / 1 8000 mph
• This is incorrect
• The correct answer can be found from 8000ft
  1.5152 miles, 1 minute 0.0167h
• So, speed 1.5152 / 0.0167 90.73 mph
• Units are important feet are not miles, minutes
  are not hours, metres are not kilometres, bits
  are not bytes, etc.

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The decimal place accuracy

• The actual speed of the last example is
• 90.730538922155688622754491017964mph
• No car speedo is that accurate so why write down
  all the figures?
• 91 mph is close enough
• The moral of this tale is give enough accuracy to
  be useful just because you can calculate
  something to 30 decimal places doesnt mean you
  have to!
• Use your judgement as a general rule two or
  three decimal places is plenty

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Common Units

• Distance metres (m), kilometres (km)
• Weight gram (g) kilogramme (km)
• Force Newton (N) kiloNewtons (kN)
• Energy Joule (J) kiloJoules (KJ)
• The above are SI units but there are many ways of
  expressing the same quantity e.g. speed in mph or
  kph
• Note thousands of a quantity are also unit
  kiloXXXXXX

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Big and small numbers
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Big and small numbers continued

• We can use scientific notation to express these
  numbers
• We write the most significant figures multiplied
  by a power of 10
• 1000 1 X 103 0.001 1 X 10 -3
• Student homework Using the last table write down
  all the powers of 10

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Rules of Powers of Ten

• Division
• 1 X 103 / 1 X 10 -5 1 X 10 8
• Invert the sign of the divider power and add to
  the numerator power
• Multiplication
• 1 X 103 X 1 X 10 -5 1 X 10 -2
• Just add the powers
• In both cases, treat the number to the left of
  the decimal point as normal numbers
• 6 X 103 / 4 X 10 -5 1.5 X 10 8

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Indices

• What you have just seen with the powers of 10 is
  an example of
• THE LAW OF INDICES
• This law is the basis of logarithms
• Historically, logs were used to aid people doing
  calculations with big or small numbers
• Now we have calculators so why bother with logs?
• Because the laws of nature have not been
  superseded by computers!!!!
• Population growth, radioactive decay, hearing and
  vision are all logarithmic functions
• The use of logs makes life easier when we are
  designing audio and other systems

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Logarithms

• Are expressed as log to base 10 or log to base
  2 or log to base X
• We can have logs in any base
• The log of 2 to base 10 is 0.3010
• This means 2 10 0.3010
• If we raise 10 to the power 0.3010 we get the
  number 2
• Remember if we want to multiply we add indices,
  divide becomes subtract

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Logarithms continued

• 2 x 2 10 0.3010 X 10 0.3010 10 0.6020
• To get the final answer we use the log tables or
  on a calculator and look up 0.6020. The answer is
  4
• 2 / 2 10 0.3010 / 10 0.3010 10 1
• To get the final answer we use the log tables or
  on a calculator and look up 1. The answer is 1
• The conversion from a log to a real number is
  called the antilog.

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Decibels

• These are logarithmic units
• Used a lot in audio and power circuit design
  because they make life easy.
• Decibels are based on power ratios and have the
  symbol dB
• Typically,
• dB 10 log 10 (Power out/Power in)
• Suppose we have an amplifier with the OP power
  twice that of the input power. What is the gain
  expressed in dB?

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Decibels continued

• dB 10 log 10 (Power out/Power in)
• 10 log 10 (2/1)
• 10 X 0.3010
• 3 dB
• 3 dB corresponds to a power GAIN of 2
• -3dB corresponds to a power LOSS of 2 i.e. half
  as much power

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Decibels continued

• 3dB is an important figure because
• It means twice as much power (or power is halved
  if 3dB)
• Our ears cannot detect changes of much less than
  3dB so a sound system that amplifies by 1 or 2 dB
  is not a lot of use!
• As we will see, decibels have many other uses.