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Lecture #1: Section 1 – Introduction and Mathematical Concepts, Part 1
One important aspect of physics, and many other sciences, is that it is quantitative.
We can take careful measurements, and then try to determine what they mean.
Measurements most often involve numbers, and so we will start off talking about what we mean when we use numbers.
In physics “2” often means “about 2”.
What is the difference between saying a distance is
2000 meters 2100 meters 2130 meters 2132 meters
These numbers have different precisions. In physics we understand these numbers to mean approximately
2000 meter 2000m ± 500m approximately within 500m to 2000m, or between 1500m and 2500m
2100 meter 2100m ± 50m approximately within 50m to 2100m, or between 2050m and 2150m
2130 meter 2130m ± 5m approximately within 5m to 2130m, or between 2125m and 2135m
2132 meter 2132m ± 0.5m approximately within 0.5m to 2132m, or between 2131.5m and 2132.5m
These digits are called significant figures.
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2000 one sig. fig. 2100 two sig. fig.’s 2130 three sig.fig.’s 2132 four sig.fig.’s
In general, how many sig.fig.’s numbers have and which ones are significant?
Non-zero digits? Just count them! 4367 four, 23.86478 seven, 2.68x10-6 3
Zeroes? Three rules!
Rule 1: Zeroes to the left of all non-zero digits are never significant.
0.0023 two 0.000000367 three 0.0000000004 one
Rule 2: Zeroes between non-zero digits are always significant. (sandwiched zeroes)
104 three 2.30046 six 5050.003 seven
Rule 3: Zeroes to the right of all non-zero digits are significant only if there is a decimal point present.
1300 two 65000. five 39.00 four 60.000 five
1.45 1.005 1.20 120 1000000 0.00004050
With large and small numbers we will often write them in scientific notation, since it’s easier, and the number of significant figures is always obvious.
578 = 1670000 = 0.00053 =
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How do we combine numbers?
Multiplication
2.23 x 5.8943 =
But is our result really this precise?
Let’s think about it…
2.23 could really be some number as small as 2.225 or as large as 2.235
5.8943 could really be some number as small as 5.89425 or as large as 5.89435 So the value of 2.23 x 5.8943 could be as small as
or as large as
So what is the “correct” answer? We expect it to be approximately between
So the best we can say is that the answer is _______________,
we don’t know for sure what the next digit is.
Notice
2.23 x 5.8943 = 13.1 __ sig. fig’s x __ sig. fig.’s = __ sig. fig.’s
A general rule of thumb is that when multiplying two numbers (or a group of numbers) the answer should have the same number of significant figures as the multiplying number with the fewest significant figures.
Division is exactly the same way.
3.0 x 5.158 = (15.474) =
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3.24 x 1.89763 = (6.1483212) = (0, 1, 2, 3, 4 rounds down 5, 6, 7, 8, 9 rounds up)
(1.67x103) / (3.2875x102) = (5.079848) =
Note, some unexpected things can happen … what is 3 x 4 =?
Be careful – your calculator does not keep track of significant figures!
How do you add numbers?
5.86x106 + 1.07x104 = ?
First, you must write them with similar exponents,
5.86x106 + 1.07x104 =
Then add the numbers without worrying about the exponents, but keeping in mind the precision of the numbers!
586 + 1.07 =
so the answer is
Subtraction is the same as addition … write the numbers with similar exponents before combining them.
In general, I will keep 3 sig. fig.’s , except when I am asking a specific significant figure question, or in certain circumstances when more than three is needed.
There is an exception for some numbers. Occasionally, in an equation we will find a factor of 2, or 4, or (1/2) or something like that. These numbers can be treated as having as many significant figures as you want. I explain these as they come up.
Also, in definitions, for example, 1 meter = 100 centimeters . These number are exact, by definition, that is 1.00000000000000 meters = 100.00000000000cm .
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Now let’s look at some measurements.
We will start with three fundamental units, one each for
length mass time
There are many possible units that we could choose, but we will use the SI (Systeme International)
Length meter, m Mass kilogram, kg Time second, s
Why is it so important to have precisely defined units? So that one person’s measurements can be compared accurately to those made by someone else.
The reason we choose the SI system (mks system) is because it is based on factors of 10 and 1000.
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SI Units
meter (m) kilogram (kg)
second (s)
1000 m = 1000 meters = 1 kilometer = 1 km = 103 m 1000 g = 1000 grams = 1kilogram = 1 kg = 103 g 1000 s = 1000 seconds = 1 kilosecond = 1 ks = 103 s
1000 km = 1 megameter = 1 Mm = 106 m 1000 Mm = 1 gigameter = 1 Gm = 109 m 1000 Gm = 1 terrameter = 1 Tm = 1012 m
(1/1000) m = 1 millimeter = 1 mm = 10-3 m , 1 m = 103 mm (1/1000) mm = 1 micrometer = 1 µm = 10-6 m , 1 m = 106 µm (1/1000) µm = 1 nanometer = 1 nm = 10-9 m , 1 m = 109 nm (1/1000) nm = 1 picometer = 1 pm = 10-12m , 1 m = 1012 pm (1/1000) pm = 1 femtometer = 1 fm = 10-15m , 1 m = 1015 fm
(1 femtometer = 1 fermi)
also
(1/100) m = 1 centimeter = 1cm = 10-2 m , 1 m = 102 cm = 100 cm
You will need to remember
(n) nano- , (µ) micro- , (m) milli- , (c) centi- , (k) kilo- , (M) mega-
