Significant Figures and Order of Magnitude

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Concept of Significant Figures:

• The digits, whose values are accurately known in a particular measurement are called significant figures.
• The number expressing the magnitude of a measurement should be recorded with only as many as digit as properly indicate the accuracy of the measurement. When the calculation is made with this numbers the final answer should contain the same number of significant figures as that in original data.
• The number of Significant figures depends upon the accuracy of the measuring instrument.
• Example: consider a measurement of the rod using metre scale of least count 0.1 cm. Let us assume that the length measured is 3.5 cm. It means that the length of the rod may be anywhere in between 3.4 cm and 3.6 cm. It means we are sure of the digit 3 but we are not sure about the digit 5. But still, the digit 5 is trustworthy. Hence there are two significant figures.
• If the length of the same rod is measured using vernier calliper of least count  0.01 cm. Let us assume that the length measured is 3.52 cm. It means that the length of the rod may be anywhere in between 3.51 cm and 3.53 cm. It means we are sure of the digit 3 and 5 but we are not sure about the digit 2. But still, the digit 2 is trustworthy. Hence there are three significant figures.
• Hence we can say that the larger the accuracy of the measuring instrument, larger will be the number of the significant figure will be larger.

Rules for Determining Significant Figures:

• One and only one uncertain digit is to be retained in a measurement. for e.g. When we record a reading as 2.65 it means that we are sure of first two digits but we are not sure about the last digit  Hence there is only one uncertain digit
• When a number is to be rounded off to a specific number of significant figures then, If number dropped is less than five, the last digit retained is left unchanged. for e.g.  If the number is 12.42 is to be rounded to three significant figures then it is rounded off as 12.4. If number dropped is equal to or greater than five, the last digit retained is increased by 1.  for e.g.  If the number is 17.49 is to be rounded to three significant figures then it is rounded off as 17.5
• All non-zero digits are significant. All zeros occurring between two non-zero digits are significant. The zeros on the left side of the number are not significant. for e.g. in the number 0.00034, the number of significant figures is 2.
• The zeros on the right side of the number are significant because they indicate the accuracy of the measurement. for e.g. in the number 3.400, the number of significant figures is 4.
• All zeros to the right of the last rightmost non-zero digits are not significant. For e.g. in 2020 there are 3 significant figures.
• All zeros to the right of the last non-zero digit are significant if they come from the measurement. For e.g. If the distance between two points is measured 2020 cm (measured to nearest centimeter) then in this number there are four significant figures.
• If the number contains more digits than the significant figures, the number should be expressed as a power of 10. for e.g. earth mass is known to be correct up to 3 significant figures, hence it is expressed as 5.98 × 1024 kg.
Sr. No. Number Significant Figures
1 6729 4
2 0.023 2
3 26.000 5
4 6.0023 5
5 2.520 x 107 4
6 0.08240 4
7 4200 2
8 4.57 x 108 3
9 91.000 5
10 0.007 1
11 2.64 x 1024 3

Rules for Calculation Using Significant Figures:

• Find the number with less number of significant figures and then retain only that number of significant figures in the other numbers and then add.for
• e.g. consider addition 7.2 + 7.83 + 6.287
• 7.2 + 7.83 + 6.287 = 21.317 (actual calculation)
• 7.2 contains less number of significant figures  i.e. 2.
• Rounding off other two numbers to two significant the answer is 21.3

Multiplication and Division:

• The operations are carried out and the result is rounded off to contain the same number of significant figures as those in the factors with less number of significant figures.for
• e.g. Consider multiplication 15.62 × 3.83  = 59. 8246 (actual calculation)
• 3.83 contains less number of significant figures  i.e. 3.
• The result  59. 8246 is rounded up to three significant figures as 59.8. Therefore the answer is 59.8.

Example – 1:

• The length, breadth, and height of a cuboid are measured as 8.24 m, 8,001 m, and 125.3 cm. Find the volume of the cuboid.
• Solution:
• l = 8,24 m, b = 8.001 m and h = 125.3 cm = 1.253 m
• Volume of cuboid = l × b × h = 8,24 m × 8.001 m × 1.253 m = 82.608084 m3.
• 8,24 contains less number of significant figures  i.e. 3.
• Thus the volume of the cuboid is 82.6  m3.

Example 2:

• 46.7 – 10.04 = 36.66 = 36.7

Example 3:

• (3.0 × 10-8) + (4.5 × 10-6) = (0.03 × 10-6) + (4.5 × 10-6) = (4.53 × 10-6) = 4.5 × 10-6

Example 4:

• (3.9 × 105) – (2.5 × 104) = (39 × 104) – (2.5 × 104) = 36.5 × 10=  3.65 × 10=3.7 x 105

Example 5:

• (4.0 x 10-4) – (2.5 x 10-6) = (400 x 10-6) – (2.5 x 10-6) = 397.5 x 10-6 =  3.975 x 10-4 = 4 x 10-4

Example 6:

• 09996 x 3.54 / 1.638 =  2.160 = 2.16

Example 7:

• 09996 x 3.52 / 1.756 =  2.004 = 2.00

Order of Magnitude:

• In the present age, the scientific study involves study from very minute particles like subatomic particles to large bodies like a star. To have a quantitative idea of universe man uses units. Thus we have to deal with very small numbers as well as with very large numbers. If these numbers are converted to the nearest power of 10 notation the comparison is easy.
• The order of magnitude of a physical quantity is defined as the value of its magnitude rounded off to the nearest integral power of 10.
• For e.g. if a man has a weight of 96 kg, then his weight can be rounded off to the nearest integral power of 10 is 100 kg or 10² kg. Hence the order of magnitude of man is 10² kg.

Order of magnitude of Length:

• In this universe, the various objects have lengths over a wide range. The following table gives the order of magnitude of their lengths from the size of the nucleus to the size of the universe.
Sr. No. Size / Distance Order of Magnitude in m
2 Radius of hydrogen atom 10-11
3 Mean free path of air molecule 10-6
4 Tip of a sharp pin 10-5
5 Thickness of sheet of paper 10-4
6 Height of man 1
7 Height of Mount Everest 104
8 Radius of the earth 107
9 Distance of the earth from sun 1011
10 Distance of nearest star from the earth 1016
11 Size of our galaxy (Milky way) 1021
12 Size of the universe 1025

Order of magnitude of Mass:

• In this universe, the various objects have masses over a wide range. The following table gives the order of magnitude of their masses from the electron to the universe.
Sr. No. Mass Order of Magnitude in kg
1 Electron 10-30
2 Proton 10-27
3 Uranium atom 10-25
4 Cell 10-10
5 Dust particle 10-9
6 man 102
7 Elephant 103
8 Ship 105
9 Moon 1023
10 Earth 1025
11  Sun 1030
12 Galaxy 1042
12 Universe 1055

Order of magnitude of Time:

• The following table gives the order of magnitude of the time intervals of few events.

Sr. No. Event Order of Magnitude in second
1 Time interval between heart beats 10-2
2 One minute 102
3 One hour 103
4 One day 105
5 One year 107
6 Average life span of human being 109
7 Time elapsed, since human life came to earth 1013
8 Age of the earth 1017
9 Age of the sun 1018

To Find Order of Magnitude:

• Find the order of magnitude of the following.
• Acceleration due to gravity = 9.8 m s-2.

Acceleration due to gravity = 9.8 m s-2 =  9.8 × 100 m s-2

Hence the order of magnitude of acceleration due to gravity is 101.

• The gravitational constant = G  = 6.67 × 10-11 N m2 kg-2

The gravitational constant = G  = 6.67 × 10-11 N m2 kg-2

Hence the order of magnitude of the gravitational constant is 10-10 N m2 kg-2.

• The charge of electron  = e  = 1.6 × 10-19 C

The charge of electron  = e  = 1.6 × 10-19 C

Hence the order of magnitude of the charge of electron is 10-19 C.

• The value of 1/4πε0  = 9 × 109 N m2 C-2.

Value of 1/4πε0  = 9 × 109 N m2 C-2.

Hence the order of magnitude of the value of 1/4πε0 is 1010 N m2 C-2.

• The mass of an electron  = 9 × 10-31 kg.

The mass of electron  = 9 × 10-31 kg.

Hence the order of magnitude of the mass of an electron is 10-30 kg.

• The radius of the earth  = 6400 km

The radius of the earth  = 6400 km = 6.4 × 103 m.

Hence the order of magnitude of the radius of the earth is 104 m.

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