1. Measurement of large distance (Parallax Method)
equation $$D=\frac{b}{\theta }$$ where D = distance of the planet from the earth.
$$\theta$$ = parallax angle.
b = distance between two place of observation.
2. Measurement of the size of a planet or a star.
equation $$\alpha =\frac{d}{D}$$ where D = distance of planet from the earth,
d = diameter of planet.
$$\alpha$$ = angular diameter of planet.
3. Measurement of mass
The gravitational force on an object, of mass m, is called the weight of the object.
1 amu = $$1.66\times 10^{27}kg$$ = 1u
4. Absolute Error - Suppose the values obtained in several measurement of physical quantitya are $$a_{1},a_{1},a_{1},....a_{n}$$ If their arithmetic mean is $$\bar{a}$$ then
$$\bar{a}=\frac{a_{1}+a_{2}+a_{3}+....+a_{n}}{n}=\frac{1}{n}\sum_{i=1}^{n}a_{i}$$
$$\Delta a_{1}=\bar{a}-a_{1}$$, $$\Delta a_{2}=\bar{a}-a_{2}$$, ........ $$\Delta a_{n}=\bar{a}-a_{n}$$
$$\Delta a_{1}, \Delta a_{2}, \Delta a_{3}.................\Delta a_{n}$$ are called absolute error.
5. Average absolute error
$$\Delta a=\frac{\left | a_{1} \right |+\left | a_{2} \right |+....+\left | a_{n} \right |}{n}=\frac{1}{n}\sum_{i=1}^{n}a_{i}$$
6. Fractional Error $$\delta _{a}=\frac{\Delta \bar{a}}{a}$$
7. Percentage Error = $$\delta _{a}\times 100$$% = $$\frac{\Delta \bar{a}}{a}\times 100$$%
8. Combination of errors
Addition $$Z=A+B\Rightarrow \Delta Z = \Delta A+\Delta B$$
Substraction $$Z=A-B\Rightarrow \Delta Z = \Delta A+\Delta B$$
Division $$Z=\frac{A}{B}\Rightarrow \frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$$
Multiplication $$Z=A\cdot B\Rightarrow \frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$$
Power $$Z=A^{n}\Rightarrow \frac{\Delta Z}{Z}=n\frac{\Delta A}{A}$$
9. Rule for determining number of significant figures
a. All the non - zero digits are significant
b. All the zeros between two non zero digits are significant no matter where the decimal point is it at all.
c. If the number is less then 1 then zeros on the right of decimal point but to the left of the first non - zero digit are not significant.
d. In a number without decimal point the zeros on the right side of the last non zero digit are not significant.
10. Dimensions and Dimensional formulas.
a. 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.
b. The power of exponents of M, L, T, K, A are called dimensions of that quantity.
11. Some important units of distance
a. 1 fermi (fm) = $$10^{-15}$$m
b. $$1\dot{A} =10^{-10}$$m
c. 1 AU = 1.496x$$10^{11}$$m
d. 1 light year = 9.46x$$10^{15}$$m
e. 1 par sec = 3.08x$$10^{16}$$m