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\(\propto\) \({1\over l}{\sqrt{T\over m}}\)

A^o= 10^-10 m
Atomic volume of 1 mole of hydrogen
= Avagadro's number\(\times\)volume of hydrogen molecule
= 6.023\(\times\)10²³ \(\times\)\(4\over 3\) \(\times\) \(\pi\) x (10^-10 m)³
= 25.2\(\times\)10^-7 m³
Molar volume = 22.4 L = 22.4\(\times\)10^-3 m³
\(Molar \ volume \over Atomic\ volume\)= 0.89x10⁴\(\approx \)104
This ratio is large because actual size of gas molecule is negligible in comparison to the inter molecular separation.

Let  T = Kr^aM^bG^c  ..... (1)
Where  K = a dimensionless constant
dimensions of the various quantities are [T] = T, [r] = L, [M] = M
[G] \(={Fr^2\over m_1m_2}={MLT^{-2}.L\over MM}=M^{-1}L^3T^{-2}\)
Substituting these dimensions in equation (1) we get,
[T] = [L]^a [M]^b [M^-1 L³ T^-2]^c
M^OL^oT^I=M^b-c=M^b-cL^a+3cT^-2c
Equating the dimensions of M, L and T, we get b - c = 0, a + 3c = 0, - 2c = 1
on solving a=\(3\over2\) b=\(-{1\over2}\) c=\(-{1\over2}\) T=Kr^3/2M^-1/2 G^-l/2 or T²=\({K^2R^3\over MG} \Rightarrow \therefore T^2 \propto r^3\)

(i) Given \(\lambda ={h\over mv}\)
As wavelength is a distance, 
LHS = \(\therefore [\lambda]=L\)
Also, RHS =\([{h\over mv}] ={Planck's \ constant \over mass \times velocity}={ML^2T^{-1}\over MLT^{-1}}=L\)
\(\therefore\) Dimensions of LHS = Dimensions of RHS.
Hence the given equation is dimensionally consistent.
(ii) Given  v = \(\sqrt{2GM\over R}\)
LHS = [V] = LT^-1
\(RHS=[{2GM\over R}]^{1\over2}=[{M^{1\over2}L^3T^{-2}.M \over L}]^{1\over 2}=[L^2T^{-2}]{1\over2}=LT^{-1}\)
\(\therefore\) Dimensions of LHS = Dimensions of RHS.
Hence the given equation is dimensionally correct.

As F \(=G{m_1m_2\over r^2}\)
\(G={F.r^2\over m_1m_2}\)
\([G]={{MLT}^{-2}.L^2\over MM}={M}^{-1}L^3{T}^{-2}\)
\(\therefore\) a = -1, b ~ 3, c = -2

\(\therefore\) \(n_2=n_1{\left[ {M_1 \over M_2} \right]}^{a}{\left[ {L_1 \over L_2} \right]}^{b}{\left[ {T_1 \over T_2} \right]}^{c}\)
\(=6.67\times{10}^{-8}{\left[ {{1\over 1000}} \right]}^{-1}{\left[ {{1\over 100}} \right]}^{3}\left[ {1\over 1} \right]^{-2}=6.67\times{10}^{}-11\)
Hence in SI units,   G = 6.67\(\times\)10^-11Nm² kg^-2

\(Average \ length ={3.48+3.46+3.49+3.50+3.48\over 5}={17.41\over5}\)
= 3.482 m = 3.48 m
(Round off to 2 places of decimal point)
The absolute errors in the different measurements are
\(\triangle\) L₁= 3.48 - 3.48 = 0.00 m
\(\triangle\) L₂= 3.48 - 3.46 = 0.02 m
\(\triangle\) L₃=3.48 - 3.49 = - 0.01 m
\(\triangle\) L₄= 3.48 - 3.50 = - 0.02 m
\(\triangle\) L₅= 3.48 - 3.48 = 0.00 m .
The absolute error =\({\sum |\triangle L_i|\over 5}\)
=\(0.00+0.02+0.01+0.02+0.00\over 5\)
=\({0.05\over5}=0.01m\)
\(\therefore\) Correct length = 3.48 ± 0.01m
Percentage error =\({0.01\over 3.48}\times 100=0.29\%\)

(i) Error in the difference of two quantities:
Let \(\triangle\)A and \(\triangle\)B be the absolute errors in the two quantities, A and B, respectively. Then,
Measured value of A = A \(\pm \triangle\)A
Measured value of B = B \(\pm \triangle\)B
Consider the difference, Z =A - B
The error \(\triangle\)Z in Z is then given by
\(Z\pm\triangle Z=(A\pm\triangle A)-(B\pm\triangle B)\)
\(=(A-B)\pm(\triangle A+\triangle B)\)
\(=Z\pm(\triangle A + \triangle B)\)
(or) \(\triangle Z=\triangle A+\triangle B\)
(ii) Error in the division or quotient of two quantities: Let \(\triangle\)A and \(\triangle\)B be the absolute errors in the two quantities A and B respectively.
Consider the quotient, Z = \(A\over B\)
The error \(\triangle\)Z in Z is given by \(Z\pm\triangle Z={A\pm\triangle A\over B\pm\triangle B}={A{(1\pm{\triangle A \over A})\over B{(\pm {\triangle B\over B})}}}={A\over B}(1\pm{\triangle A\over A})(1\pm{\triangle B\over B})^{-1}\)
\(or Z\pm \triangle Z=Z(1\pm {\triangle A\over A})(1\mp{\triangle B\over B}) \)        [using (1 +x)ⁿ=1+ nx, when x «1]   
Dividing both sides by Z, we get, \(1\pm{\triangle Z\over Z}=(1\pm{\triangle A\over A})(1\mp{\triangle B\over B})=1\pm {\triangle A\over A}\mp {\triangle B\over B}\pm{\triangle A\over A}{\triangle B\over B}\)
As the terms \(\triangle\)A / A and \(\triangle\)B/B are small, their product term can be neglected,
The maximum fractional error in Z is given by \({\triangle Z \over Z}=({\triangle A \over A}+{\triangle B\over B})\)
(iii) Error in the power of a quantity: Consider the n^th power of A, Z = Aⁿ The error\(\triangle\) Zin Z is given by
Z\(\pm \triangle\)Z=(A\(\pm \triangle\)A)ⁿ=Aⁿ =\(=(1\pm{\triangle A\over A})^n=Z(1\pm n{\triangle A\over A})\)
We get [(1+x)ⁿ + nx, when x« 1] neglecting remaining terms, Dividing both sides by Z
\(1\pm{\triangle Z \over Z}=1\pm n{\triangle A \over A}or{\triangle Z \over Z}=n{\triangle A \over A}\)

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.

C is the centre of the Earth A and B are two diametrically opposite places on the surface of the Earth. From A and B, the parallaxes \(\theta _1\) and \(\theta _2\) respectively of Moon M with respect to some distant star are determined with the help of an astronomical telescope.
Thus, the total parallax of the Moon subtended on Earth \(\angle\)AMB = \(\theta _1\)+ \(\theta _2\) =\(\theta \)

If \(\theta\) is measured in radians, then\(\theta ={AB\over MC};AM\approx MC\)
\(\theta ={AB\over MC} or M.C={AB\over\theta}\)
Knowing the values of AB and \(\theta\) , we can calculate the distance MC of Moon from the Earth. 

Errors in the difference of two quantities.
Let \(\triangle A\) and \(\triangle B\) be the absolute errors in the two quantities, A and B, respectively. Then,
Measured value of  \(A=A\pm\triangle A\)
Measured value of \(B=B\pm\triangle B\)
Consider the difference, Z =A - B
The error \(\triangle Z\) in Z is the given by
\(Z\pm \triangle Z=(A+\triangle A)-(B\pm \triangle B)\)
\(=(A-B)\pm(\triangle A+\triangle B)\)
\(=Z\pm(\triangle A+\triangle B)\)
(or) \(\triangle Z=\triangle A+\triangle B\) 
The maximum error in difference of two quantities is equal to the sum of the absolute errors in the individual quantities. Error in the division or quotient of two quantities
Let \(\triangle A\) and \(\triangle B\) be the absolute errors in the two quantities A and B respectively.
Consider the quotient, \(Z={{A}\over{B}}\)
The error \(\triangle Z\) in Z is given by
\(Z\pm Z={{A\pm \triangle A}\over{B+\triangle B}}={{A\left(1\pm{{{\triangle A}\over{A}}} \right)}\over{B\left( 1\pm{{\triangle B}\over{B}} \right)}}\)
\(={{A}\over{B}} \left( 1\pm{{\triangle A}\over{A}} \right)\left( 1\pm{{\triangle B}\over{B}} \right)^{-1}\)
or \(Z\pm \triangle Z=Z\left( 1\pm{{\triangle A}\over{A}} \right)\left( 1\mp{{\triangle B}\over{B}} \right)\)
[ using (1+x)n \(\approx\) 1 + nx, when x<<1]
Dividing both sides by Z, we get
\(1\pm{{\triangle Z}\over{Z}}=\left( 1\pm{{\triangle A}\over{A}} \right)\left( 1\mp {{\triangle B}\over{B}} \right)\)
\(=1\pm{{\triangle A}\over{A}}\mp{{\triangle B}\over{B}}\pm{{\triangle A}\over{A}}.{{\triangle B}\over{B}}\)
As the terms \(\triangle A/A\) and \(\triangle B/B\) are small, their product term can be neglected.
The maximum fractional error in Z is given by
\({{\triangle Z}\over{Z}}=\left( {{\triangle A}\over{A}} +{{\triangle B}\over{B}}\right)\)
The maximum fractional error in the quotient of two quantities is equal to the sum of their individual fractional errors.