1.1 – Measurements in physics
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Scientific notation
One of the fascinations for physicists is dealing with the very large (e.g. the universe) and the very small (e.g. electrons). Many physical constants (quantities that do not change) are also very large or very small. This presents a problem: how can writing many digits be avoided? The answer is to use scientific notation.
The speed of light has a value of 299 792 458 m s^−1. This can be rounded to three significant figures as 300 000 000 m s^−1. There are a lot of zeros in this and it would be easy to miss one out or add another. In scientific notation this number is written as 3.00 × 10^8 m s^−1 (to three significant figures).
Let us analyse writing another large number in scientific notation. The mass of the Sun to four significant figures is 1 989 000 000 000 000 000 000 000 000 000 kg (that is 1989 and twenty-seven zeros). To convert this into scientific notation we write it as 1.989 and then we imagine moving the decimal point 30 places to the left (remember we can write as many trailing zeros as we like to a decimal number without changing it). This brings our number back to the original number and so it gives the mass of the Sun as 1.989 × 10^30 kg.
Apart from avoiding making mistakes, there is a second reason why scientific notation is preferable to writing numbers in longhand. This is when we are dealing with several numbers in an equation. In writing the value of the speed of light as 3.00 × 10^8 m s^−1, 3.00 is called the “coefficient” of the number and it will always be a number between 1 and 10. The 10 is called the “base” and the 8 is the “exponent”.
There are some simple rules to apply:
- When adding or subtracting numbers the exponent must be the same or made to be the same.
- When multiplying numbers we add the exponents.
- When dividing numbers we subtract one exponent from the other.
- When raising a number to a power we raise the coefficient to the power and multiply the exponent by the power.
Metric multipliers (prefixes)
Scientists have a second way of abbreviating units: by using metric multipliers (usually called “prefixes”). An SI prefix is a name or associated symbol that is written before a unit to indicate the appropriate power of 10. So instead of writing 2.5 × 10^12 J we could alternatively write this as 2.5 TJ (terajoule). Check this following figure.
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| http://www.icrf.nl/Portals/106/SI_prefixes.jpg |
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