System of Units
Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units.
International System of Units
In earlier times, Scientists of different countries were using different systems of units for measurement. The system of units which is at present internationally accepted for measurement is International System of Units, abbreviated as SI. In SI, there are seven base units:
|Base Quantity||SI Unit Name||Symbol|
|Amount of Substance||mole||mol|
Errors in Measurement
The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.
Accuracy and Precision
The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. For example, suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise.
The smallest value that can be measured by the measuring instrument is called its least count. All the readings or measured values are good only up to this value.
Estimation of Error
Absolute Error: Suppose the values obtained in several measurement of physical quantity a are a1, a2, ...... an. The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. In absence of any other method of knowing true value, arithmetic mean is considered as the true value.
The arithmetic mean of all the absolute errors is taken as the final or mean absolute error.
Relative Error: The relative error is the ratio of the mean absolute error to the mean value of the quantity measured.
Percentage Error: When the relative error is expressed in percent, it is called the percentage error.
Combination of Errors
If an experiment is done involving several measurements, the errors in all the measurements combine. For example, density is the ratio of the mass to the volume of the substance. If there are errors in the measurement of mass and of the sizes, then use combination of errors to determine the error in the density of the substance.
The reliable digits plus the first uncertain digit are known as significant digits or significant figures.
Number of Significant Figures
- All the non - zero digits are significant.
- All the zeros between two non zero digits are significant no matter where the decimal point is.
- If the number is less than 1, then zeros on the right of decimal point but to the left of the first non-zero digit are not significant.
- In a number without decimal point the zeros on the right side of the last non-zero digit are not significant.
Dimensions and Dimensional Formulas
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. The expression of a physical quantity with appropriate powers of M, L, T, K, A, etc is called the dimensional formula of that physical quantity. The power of exponents of M, L, T, K, A are called dimensions of that quantity.
For example, the dimensions of force are [M L T-2].
Checking the Dimensional Consistency of Equations
The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. Velocity cannot be added to force, or an electric current cannot be subtracted from the thermodynamic temperature. This principle called the principle of homogeneity of dimensions.
Deducing Relation among the Physical Quantities
The method of dimensions can be used to deduce relation among the physical quantities. However, dimensionless constants cannot be obtained by this method.