1 Radian (rad): One radian is the angle subtended at the center of a circle
by an arc equal in length to the radius of the circle. \[d\theta =\frac{ds}{r}\]SI unit is radian.
2. Steradian (sr): One steradian is the solid angle subtended at the center of a
sphere, by that surface of the sphere, which is equal in area, to the square of the radius of the sphere. \[d\Omega = \frac{dA}{r^2}\]SI unit is steradian
3. Astronomical Unit (AU): It is the average distance of the centre of the sun from the
centre of the earth. 1 AU =1.496 x 1011 m =1.5 x 1011 m
4. Light year (ly): One light year is the distance travelled by light in vacuum in one
year. 1 ly = 9.46 x1015m.
5. Parsec : One parsec is the radius of the circle at the centre of which an
arc of the circle, 1 AU long subtends an angle of 1”. 1 parsec = 3.1 x 1016 m
6. Relation between AU, ly and par sec, 1 ly = 6.3 x 104 AU, 1 par sec = 3.26 ly
7. In the micro-cosm measurement,
(i)
1 micron = 1 μ or 1 μm = 10-6 m
(ii)
1 nanometer = 1 nm =10-9 m
(iii)
1 angstrom = 1A0 = 10-10
m
(iv)
1 fermi = 1 femtometer = 1 fm = 10-15
m
8. For measuring very small area,
1 acre = 4047 m2
1 are (a) = 102 m2
1 hactare = 104 m2
9. For measuring heavy masses,
(i)
1 tonne or 1 metric ton = 1000 kg
(ii)
1 quintal = 100 kg
(iii)
1 slug = 14.57 kg
(iv)
1 lb = 0.4536 kg
10. For measuring very small masses, 1 atomic mass unit = 1 a.m.u. or 1 u = 1.66 x 10-27 kg.
11. Some practical units of standard of time are :
(i)Solar day: It is the time interval between two
successive passage of the sun across the meridian.
(ii) Sedrial day: It is the time interval between
two successive passages of a fixed star across the meridian.
(iii) Solar year (or year) is the time taken by the
earth to complete one revolution around the sun in its orbit.
1 solar
year = 365.25 average solar days = 366.25 sedrial days
The year in which there is total solar eclipse is called a
tropical year. The year which is divisible by 4, and I which month of February
has 29 days, is called a leap year. One hundred years make up 1 century.
(iv) Lunar month. It is the time taken by moon to
complete one revolution around the earth in its orbit. 1 Lunar month = 27.3 days.
(v) Shake: It s the smallest practical unit of
time. 1 shake = 108s
12. Parallax method: Parallax is the name of the name given to change in the position
of an object with respect to the background when the object is seen from two
different positions. The distance between the two-position (i.e., points of
observation) is called the basis. \[\Theta = \frac{b}{x}\] where b is the arc length, x is the radius and $ \Theta $ is angle subtended. The parallax method has been used for measuring the distance of stars
of which are less than a hundred light-years away.
13.
Error: The difference between the true value and the measured value of any physical quantity is called error. i.e. Error = [True Value] - [Measured Value].
14. There are three types of errors, namely, Systematic, Random and gross error.
15. Systematic Error: There error tend to be in any one direction either positive or negative. some of the systematic errors are: Instrumental error, Imperfection in experimental technique or procedure, Personal error, Least count error.
16. Random Error: These errors occur irregularly. It arises due to random and unpredictable variation in experimental conditions like Temperature, Pressure, voltage, etc. It can be minimized by repeating the experiments.
17. Gross Error: These errors arise due to the carelessness of the observer. For example, Reading an instrument improperly, noting observations incorrectly, using wrong values in the calculation, etc.
18. Absolute Error: It is the magnitude of the difference between the true value and the individual measured value of the quantity. Let physical quantity be measured n times and observed values be $ a_1, a_2,........,a_n $. Then, arithmetic mean of these value are, \[a_m = \frac{a_1+a_2+.....+a_n}{n} \implies a_m =\frac{1}{n}\Sigma_{i=1}^{i=n}a_i\]Then absolute error in any measured value is given by \[\Delta a_i = a_m - a_i\]
19. Means absolute error: It is arithmetic mean of the magnitude of absolute errors in all measurements of quantity. It is represented by $\Delta a_{mean} $. Thus, \[\Delta a_{mean}=\frac{|\Delta a_1|+|\Delta a_2|+.....+|\Delta a_n|}{n} \implies \Delta a_{mean} = \frac{1}{n}\Sigma_{i=1}^{i=n}|\Delta a_i |\]Hence final result of measurement may be written as
\[a=a_m \pm \Delta a_{mean}\]
20. Relative Error: It is defined as the ratio of mean absolute error to the mean value of the quantity measured. Thus, \[\delta a = \frac{\Delta a_{mean}}{a_m}\]
21. Error in Sum: Let x = a + b, then maximum absolute error in x is \[\Delta x = \pm (\Delta a +\Delta b)\]Hence maximum absolute error in sum of two quantities is equal to sum of the absolute errors in the individual quantities.
22. Error in difference: Let x = a - b, then maximum absolute error in x is \[\Delta x = \pm (\Delta a +\Delta b)\] Hence maximum absolute error in difference of two quantities is equal to sum of the absolute errors in the individual quantities.
23. Error in Product: Let $ x $ = a x b, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm (\frac{\Delta a}{a} +\frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities.
24. Error in Product: Let $ x = \frac{a}{b} $, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm (\frac{\Delta a}{a} +\frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities.
25. Error in case of measured quantity raised to a power: Let $ x = \frac{a_n}{b_m} $, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm ( n \frac{\Delta a}{a} + m \frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities.
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