In class XI physics course, the topic of "Physical Quantities and Their Measurements" is a useful introduction. A physical quantity is measured in terms of a small part of it. The small part is conventionally adopted as a unit of measurement of the quantity. It is helpful to first establish the units of a few quantities which are called base quantities or fundamental quantities. The corresponding units are called base units or fundamental units. The units of the remaining physical quantities are expressed in terms of these base units. These quantities are called derived quantities, and their units are called derived units. The set of base and derived units is called a system of units.In 1960, an international committee called CGPM adopted the International System of Units or SI which has been supplemented and refined ever since. The group of SI units consists of seven base units and two supplementary units.
The concept of order of magnitude is often very helpful to compare the magnitudes of two or more physical quantities of the same type. At first the absolute value of a given quantity is written in the form
x × 10y where
1≤ x< 10 and y is a positive or negative integer. Next, the number is rounded off according to specific guidelines. The power of 10 in the final representation is called the order of magnitude of the quantity. For instance, the mass of the earth is
5.98 × 1024 kg and the mass of an electron is about
9.11 × 10-31 kg.
By the above rules, the mass of the earth is of the order of
1025kg and the mass of an electron is of
the order of
10-30 kg .
Therefore, the mass of the earth is
25 - (-30) = 55 orders of magnitude larger than the mass of an electron.
The word dimension in physics means the physical nature of the quantity.
The dimensions of a quantity should not be confused with its units. To illustrate the point,
the distance between two places can be measured in different units such as metre, centimetre, foot etc. But the dimension of
distance is always the same, which is "length". The usual practice is to use square brackets, [ ], to denote the dimensions of a
physical quantity. Thus, the dimension of distance is
[L] , the dimension of mass is
the dimensions of velocity and force are
Any equation that relate several physical quantities must be homogenous from the dimensional point of view. If an equation is of the form A = B + C, then A, B and C must represent the same physical quantity with the same dimensions. This property is known as the principle of homogeneity of dimensions. To illustrate the point, let us take an important equation from Kinematics:
x - x0 = ut + ½ at2
(x-x0) is the distance travelled by a particle along a straight line in time t, u is the initial velocity of
the particle and a is the constant acceleration with which the particle is moving.
The dimension of
(x - x0) is [L], the dimension of ut is
[LT-1][T] = [L],
and the dimension of
at2 is [LT-2][T2] = [L]. Therefore,
the dimensional form of the equation can be written as:
[L] = [L] + [L]
which demonstrates that the equation is dimensionally homogenous.
The method of dimensional analysis has many uses:
Each one of them will be discussed with examples in due course.
We can never measure a physical quantity with absolute precision. The degree of accuracy of any measurement depends on the smallest scale division of the measuring instrument. This is known as the least count of the instrument. The least count of an ordinary metre scale is 0.1 cm or 1 mm. When we use it to measure the length of a certain steel rod, suppose the reading falls somewhere between 24.6 cm and 24.7 cm marks. So we divide the space between the two marks mentally in 10 equal parts and guess how many of these parts are the from the 24.6 cm mark to the end of the rod. Let this number be 3. Accordingly, the length of the rod is taken as 24.63 cm. So we see here that the first three digits counted from the left, i.e. 2, 4 and 6 are certain but the fourth digit 3 is uncertain. The significant figures of any measurement are the digits which are reasonably reliable. By convention, they consist of all the certain digits and only one uncertain digit. Thus, a metre scale can read up to four significant figures (2, 4, 6 and 3 in our example).
Certain points should be kept in mind while counting significant figures in a number. Certain rules are to be followed in counting significant figures in the answer of algebraic operations such as multiplication, division, addition and subtraction. They will be discussed in due course.
Any experiment involves a series of measurements. Several errors may occur during these measurements. The errors in measurement can be classified into two basic types: systematic error and random error.
Often we wish to find the value of a quantity Z that involves the measurement of two other quantities X and Y.
The combined error in the value of Z depends not only on the errors in the values of X and Y, but also on the nature of
dependence of Z on X and Y. There are specific formulas which we can develop to calculate the
combination of errors for various cases such as
Z = X + Y,
Z = X - Y,
Z = XY,
Z = X / Y,
Z = Xn,
or a more general type
U = k Xa Yb / Zc.