Units and Measurement (Physical Quantities) | Notes |Problem Solutions | Class 11 (Physics) | Physics in Depth

Physics is the mathematical formulation of the natural laws. To describe the laws of nature, we need the quantitative description that involves measurement and comparison of physical quantities. To measure a physical quantity we need some standard unit of that quantity. We measure each physical quantity in its own units, by comparison with a standard . The unit is a unique name that we assign to measure of that quantity - for example, meter (m) for the quantity length. The standard corresponds to 1.0 unit of the quantity.

• The dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. [M], [L], [T], [A], etc. are the respective dimensions of mass, length, time and current.
• Homogeneity of dimensions in an equation: The dimensions of all term in an equation must be identical.i.e. dimension in L.H.S. of an equation is equal to the dimension in R.H.S. of an equation.
• Limitation of the dimensional method - a dimensionally correct equation need not be actually correct but a dimensionally wrong equation must be wrong: Consider the equation,
  \[x=ut+\frac{1}{2}at^2\] here x is the distance travelled by the particle in time t which starts at a speed u and has an acceleration a along the direction of motion. \[\begin{align*} x&=[L]\\ ut&=\text{velocity}\times \text{time}=\frac{\text{length}}{\text{time}} \times \text{time}=[L]\\ \frac{1}{2}at^2&= \text{acceleration}\times {\text{time}}^2=[L]\\ \end{align*}\] Thus the equation is correct as far as the dimension are concerned. Here the dimension of \(\frac{1}{2}at^2\) is same as that of a\(t^2\). Pure numbers(think!) are dimensionless. Dimension doesnot depend on the magnitude. Due to this reason, the equation x = ut + at\(^2\) is also dimensionally correct. Thus, a dimensionally correct equation need not be actually correct but a dimensionally wrong equation must be wrong.
• Accuracy: Accuracy is the measure of how close you are to the actual value. It depends on the measuring of a person.
  Precision: Precision is the measure of how close your measurements are to each other. It depends on the measuring tool and is determined by the number of significant digits. Think of shooting a ball into the target. Accuracy is represented by hitting the bulls eye( the accepted value) and precision is represented by a tight grouping of shots.

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• Significant figures: The significant figures of the measure of a physical quantity are all those digits about which we are absolutely sure plus one digit that has a little doubt or uncertainty. For example, we find that one end of an object coincides with the zero of the metre scale and the other end may fall between 10.4 and 10.5 cm mark of the scale. We mentally divide the 1 mm division in 10 equal parts and guess on which part is the edge falling. We may note down the reading as 10.46 cm. The digits 1, 0 and 4 are certain but 6 is doubtful. All these digits are called significant digits. We say that length is measured upto 4 significant digits. The right most or the doubtful digit is least significant and leftmost is most significant digit.
  
  1. All the non-zero digits are significant. In 2.738, the number of significant figures is 4.
  2. All the zeroes between two non-zero digits are significant, no mat- ter where the decimal point is, if at all. As examples, 209 and 3.002 have 3 and 4 significant figures respectively.
  3. If the measurement of number is less than 1, the zeroes on the right of decimal point and to the left of the first non-zero digit are non-significant. In 0.00807, red numbers are non significant and the number of significant figures is only 3.
  4. The terminal or trailing zeroes in a number without a decimal point are not significant. Thus 12.3=1230=12300 mm has only 3 significant figures.
  5. The trailing zeroes in number with a decimal point are significant. Thus, 3.800 kg has 4 significant figures.
  6. A choice of change of units does not change the number of signif- icant digits or figures in a measurement.

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