Uncertainty In Measurement - Practice Questions & MCQ

Scientific Notation

For denoting numbers as large as 602,200,000,000,000,000,000,000 for the molecules of 2 g of hydrogen gas or as small as 0.00000000000000000000000166 g mass of an H atom, we need to use scientific notation i.e., exponential notation in which any number can be represented in the form of N × 10ⁿ where n is an exponent having positive or negative values and N is a number (called digit term) which varies between 1.000... and 9.999.... Thus, we can write 232.609 as 2.32609 ×10² in scientific notation.

 

Addition and Subtraction

For adding or subtracting two numbers, first, the numbers are written such that they have the same exponent.

After that, the digit terms are added or subtracted.

4.5 X 10⁴ + 2.5 X 10⁵ = 4.5 X 10⁴ + 25.0 X 10⁴ = 29.5 X 10⁴ 

4.5 X 10⁴ - 2.5 X 10⁵ = 4.5 X 10⁴ - 25.0 X 10⁴ = (-)20.5 X 10⁴  (negative)

 

Multiplication and Division

For multiplying or dividing two numbers, the same rules are followed as for exponential numbers.

(4.5 X 10⁴) X (2.5 X 10⁵) =  (4.5 X 2.5)X (10⁴⁺⁵) = 11.25 X 10⁹ = 1.125 X 10¹⁰ 

(4.5 X 10⁴ ) / (2.5 X 10⁵) = (4.5 / 2.5) X 10^4-5 = 1.8 X 10^-1

Experimental calculations or Measurements

Precision refers to the closeness of various measurements for the same quantity.

Accuracy is the agreement of a particular value to the true value of the result.

For example, if the true value for a result is 2.00 g and student ‘A’ takes two measurements and reports the results as 1.95 g and 1.93 g. These values are precise as they are close to each other but are not accurate. Another student ‘B’ repeats the experiment and obtains 1.94 g and 2.05 g as the results for two measurements. These observations are neither precise nor accurate. When the third student ‘C’ repeats these measurements and reports 2.01 g and 1.99 g as the result, these values are both precise and accurate.

Significant Figures

Every experimental measurement has some amount of uncertainty associated with it because of limitation of measuring instrument and the skill of the person making the measurement. And hence, the result of any experiment is written indicating the significant figures which includes all the digits known with certainity plus one which is uncertain

There are certain rules for determining the number of significant figures. These are stated below:

(1) All non-zero digits are significant.

(2) Zeros preceding to first non-zero digit are not significant. Such zero indicates the position of the decimal point.

(3) Zeros between two non-zero digits are significant.

(4) Zeros at the end or right of a number are significant provided they are on the right side of the decimal point. 

(5) Counting numbers of objects, for example, 2 balls or 20 eggs, have infinite significant figures as these are exact numbers and can be represented by writing an infinite number of zeros after placing a decimal i.e., 2 = 2.000000 or 20 = 20.000000

In numbers written in scientific notation, all digits are significant e.g., 4.01×10² has three significant figures, and 8.256 × 10^–3 has four significant figures.

Addition and Subtraction of Significant Figures
The result cannot have more digits to the right of the decimal point than either of the original numbers.

  12.11
  18.0
  1.012    

  31.122

Here, 18.0 has only one digit after the decimal point and the result should be reported only up to one digit after the decimal point, which is 31.1.

Multiplication and Division of Significant Figures
In these operations, the result must be reported with no more significant figures as in the measurement with the few significant figures.

2.5×1.25 = 3.125
Since 2.5 has two significant figures, the result should not have more than two significant figures, thus, it is 3.1.

Rounding off the numbers
While limiting the result to the required number of significant figures as done in the above mathematical operation.the following points for rounding off the numbers.

1. If the rightmost digit to be removed is more than 5, the preceding number is increased by one. For example, 1.386. If we have to remove 6, we have to round it to 1.39.
2. If the rightmost digit to be removed is less than 5, the preceding number is not changed. For example, 4.334 if 4 is to be removed, then the result is rounded upto 4.33.
3. If the rightmost digit to be removed is 5, then the preceding number is not changed if it is an even number but it is increased by one if it is an odd number. For example, if 6.35 is to be rounded by removing 5, we have to increase 3 to 4 giving 6.4 as the result. However, if 6.25 is to be rounded off it is rounded off to 6.2.

Dimensional Analysis

While calculating, the units should be the same all across the equation. The units for each physical quantity should be the same throughout the equation. 

eg. If the temperature is involved, the units should be Kelvin on both sides