Dimension of Planck's constant is \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]\)
Dimension of Gravitational constant is \(\left[\mathrm{M}^{-1} \mathrm{~L}^{+3} \mathrm{~T}^{-1}\right]\)
Dimension of Velocity constant is LT^-1
Dimension of Length is L
\(\therefore \text { Dimension of } \frac{\sqrt{h G}}{C^{\frac{3}{2}}}\)
\(=\frac{\sqrt{\left(\mathrm{ML}^{2} \mathrm{~T}^{-1}\right)\left(\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right)}}{\left(\mathrm{LT}^{-1}\right)^{3 / 2}} \)
\(=\frac{\sqrt{\mathrm{L}^{5} \mathrm{~T}^{-3}}}{\mathrm{~L}^{3 / 2} \mathrm{~T}^{-3 / 2}} \)
\(=\frac{\mathrm{L}^{5 / 2} \mathrm{~T}^{-3 / 2}}{\mathrm{~L}^{3 / 2} \mathrm{~T}^{-3 / 2}} \)
\(=\mathrm{L}^{5 / 2-3 / 2} \mathrm{~T}^{3 / 2+3 / 2}=\mathrm{L}^{1} \mathrm{~T}^{0}=\mathrm{L}\)
Dimension of length = L

\(\text { Gravitational constant } G=\frac{F r^{2}}{m_{l} m_{2}}\)
\(\text { Dimensional formula of } \mathrm{G}=\frac{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^{2}\right]}{[\mathrm{M}][\mathrm{M}]}\)
\(=\frac{\mathrm{ML}^{3} \mathrm{~T}^{-2}}{\mathrm{M}^{2}}=\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\)

Dimensional formula of Planck's
\(\text { constant }=\frac{\text { Energy }}{\text { Frequency }}=\frac{\mathrm{ML}^{2} \mathrm{~T}^{-2}}{\mathrm{~T}^{-1}}\)
\(=M L^{2} T^{-2+1}=M L^{2} T^{-1}\)

\(\pi=3.14 \)
\(\pi^{2} =3.14 \times 3.14 \)
\(=9.8596=9.86 \)

248000

\(a={3km\over h^2}={3\times 10^5cm\over (3600s)^2}\)
\(={300000\over 12960000}=0.0231cms^{-2}\)

The scientific method is a step-by-step approach in studying natural phenomena and establishing laws which govern these phenomena.

(i) Barn,1 barn 10^-28m²
(ii) Acre,1 acre = 4047 m²
(iii) Hectare,1 hectare = 10⁴ m².

S_n= Distance travelled in nth see that is (S_n-S_n-1)
S_n= u\(\times\)1+\(\frac { a }{ 2 } \)[2n-1]
[LT^-1]=[LT^-1]+[LT^-2][T]
[LT^-1] = [LT^-1]
L.H.S = R.H.S
Hence this is dimensionally correct.

Let us assume that the Energy E depends on mass m and velocity of light c.
\(E\alpha m^ac^b\)
\(E=km^ac^b\) where K a constant
Dimensions of E = [ML²T^-2]
Dimensions of m = [M]
Dimensions ofc = [LT^-1]
Substituting the values in the above equation
[ML²T^-2] = K[M]^a [LT^-1]^b
By equating the dimensions
a = 1
b = 2
-b = -2
E = k.mc²
The value of constant k = 1
E = mc². This is Einstein's mass energy relation.

Angle \(\theta\) = 1° 55'= 115' = (115\(\times\)60)"\(\times\) (4.85\(\times\) 10^-6) rad = 3.34\(\times\)10^-2 rad
Since 1" = 4.85\(\times\)10^-6 rad
Radius of the Earth = 6.4\(\times\)10⁶m
From the Figure AB is the diameter of the Earth (b) = 2\(\times\)6.4\(\times\)10⁶m
Distance of the Moon from the Earth x = ?
\(x={b\over \theta}={2\times 6.4 \times 10^6\over 3.34\times 10^{-2}}\)
x = 3.83\(\times\)10⁸m

Addition and subtraction: In addition and subtraction, the final result should retain as many decimal places as there are in the number with the smallest number of decimal places.
Eg:
(i) 3.1 + 1.780 + 2.046 = 6.926
Here the least number of significant digits after the decimal is one. Hence the result will be 6.9.
(ii) 12.637 - 2.42 = 10.217
Here the least number of significant digits after the decimal is two. Hence the result will be 10.22

By the principle of homogeneity, a / V² is of the dimensions of pressure and b is of the dimensions of volume.
[a] = [pressure] [V²] = [ML⁻¹T⁻²] [L⁶]
= [ML⁵T^-2]
[b] = [V] = L³

(i) Basic laws of electricity and magnetism led to the discovery of wireless communication technology which has shrunk the world with effective communication over large distances.
(ii) The launching of satellite into space has revolutionized the concept of communication.
(iii) Microelectronics, lasers, computers, superconductivity and nuclear energy have comprehensively changed the thinking and living style of human beings.

n\(\propto\) I^aT^bm^c, [I] = [M^oL¹T^o]
[T] = [M¹L¹T^-2] (force)
[M] = [M¹L^-1T^o]
[M^o L^oT^-1] = [M^oL¹T^o]^a [M¹L¹T^-2]^b [M^oL^-1T^o]^C
b + c = 0
a + b - c = 0
-2b = -1 \(\Rightarrow\) b = \(1\over2\)
c =\(-{1\over2}a=1\)
n\(\propto\) \({1\over l}{\sqrt{T\over m}}\)

\((C+\triangle C)=(3.0\pm0.1)\mu F\)
\((V+\triangle V)=(18\pm 0.4)V\)
Q = CV
\(Q=3.0\times10^{-6}\times18=54\times10^{-6}\) coulomb
\(\text {Error in } C=\frac{\Delta C}{C} \times 100=\frac{0.1}{3} \times 100=3.3 \%\)
Error in \(V=\frac{\triangle V}{V}\times100=\frac{0.4}{18}\times100=2.2\%\)
Error in Q = Error in C + Error in V
= 3.3% + 2.2% = 5.5%
\(\therefore\) Charge Q = (54\(\times\)10⁶ 土 5.5%)C

The uncertainty in a measurement is called an error. The three possible errors are 
(i) Systematic error
(ii) Random error and
(iii )Gross error
(i) Systematic Errors: Systematic errors are reproducible inaccuracies that are consistently in the same direction. These occur often due to a problem that persists throughout the experiment. Systematic errors can be classified as follows,
Instrumental errors: When an instrument is not calibrated properly at the time of manufacture, instrumental errors may arise. If a measurement is made with a meter scale whose end is worn out, the result obtained will have errors. These errors can be corrected by choosing the instrument carefully.
Imperfections in experimental technique or procedure: These errors arise due to the limitations iri the experimental arrangement. As an example, while performing experiments with a calorimeter, if there is no proper insulation, there will be radiation losses. This results in errors and to overcome these, necessary correction has to be applied.
Personal errors: These errors are due to, individuals performing the experiment, may be due to incorrect initial setting up of the experiment or carelessness of the individual making the observation due to improper precautions. Errors due to external causes: The change in the external conditions during an experiment can cause error in measurement. For example, changes in temperature, humidity, or pressure during measurements may affect-the result of the measurement.
Least count error: Least count is the smallest value that can be measured by the measuring instrument, and the error due to this measurement is least count error. The instrument's resolution hence is the cause of this error. Least count error can be reduced by using a high precision instrument for the measurement.
(ii) Random errors: Random errors may arise due to random and unpredictable variations in experimental conditions like pressure, temperature, voltage supply etc. Errors may also be due to personal errors by the observer who performs the experiment. Random errors are sometimes called "chance error". When different readings are obtained by a person every time he repeats the experiment, personal error occurs. For example, consider the case of the thickness of a wire measured using a screw .gauge. The readings taken may be different for different trials. In this case, a large number of measurements are made and then the arithmetic mean is taken.
If n number of trial readings are taken in an experiment, and the readings are a₁,a₂,a₃,..... a_n. The arithmetic mean is 
\(a_m={a_1+a_2+a_3+...a_n\over n}(or)a_m={1\over n}\sum _{ i=1 }^{ i=n }{ { a }_{ i } } \)
Usually this arithmetic mean is taken as the best way to minimize the error.
(iii) Gross Error: The error caused clue to the shear carelessness of an observer is called gross error.
for example
(a) Reading an instrument without setting it properly.
(b) Taking observations in a wrong manner without bothering about the sources of errors and the precautions.
(c) Recording wrong observations.
(d) Using wrong values of the observations in calculations.
These errors can. be minimized only when an observer is careful and mentally alert.