The elastic spring force arise due to the net attraction/repulsion between the neighboring atoms of the spring when the spring is elongated/compressed.

This net attraction/ repulsion can be traced to the sum of electric forces between the charged constituents of the atoms.

By Avogadro's hypothesis, the actual volume occupied by the atoms in one gram of a substance is 2/3rd of the volume occupied by 1 gram of the substance.

Let us consider a sample of a substance of mass m and volume V. If M is its molecular weight and Avogadro number is N, then

Number of atoms in the given sample = {tex}{N\over M}× m = {Nm\over M}{/tex}

let each atom is a sphere of radius r..

So 

Actual volume of the atoms in the given sample = {tex}{Nm\over M}×{4\over 3}\pi r^3{/tex}

 By Avogadro's hypothesis

{tex}{Nm\over M}×{4\over 3}\pi r^3 = {2\over 3}V{/tex}

If {tex}\rho{/tex} is density of substance . Then

{tex}\rho = {m\over V} \\=> V = {m\over \rho}{/tex}

{tex}=> {Nm\over M}×{4\over 3}\pi r^3 = {2\over 3}{m\over \rho}{/tex}

{tex}r = \left ({M\over 2 \pi N\rho }\right )^{1\over 3}{/tex}

Knowing values of M,N and {tex}\rho{/tex}

We can find radius 

ofcourse the above equation is dimensionally correct

[F][S]=[m][v]²-[m][u]²

[MLT^-2][L]=[M][LT^-1]²-[M][LT^-1]²

[ML²T^-2]=[ML²T^-2]+ [ML²T^-2]

WE know subraction of same dimensions givse the same dimension

so

[ML²T^-2]=[ML²T^-2]

L.H.S.=R.H.S

hence the above equation is dimensionally correct.

Mebsion its measures.

The centripetal force, F acting on a particle moving uniformly in a circle depend upon the mass (m), velocity (v) and radius (r) of the circle.

The formula for F using the method of dimensions.

Let F = k m^a v^b r^c…..... (i)

Where, k is the dimensionless constant of proportionality, and a, b, c are the powers of m, v, r respectively.

On writing the dimensions of various quantities in (i),

we get

[M¹L¹T^-2] = M^a [LT^-1]^bL^c

=> [M¹L¹T^-2] = M^aL^bL^cT^-b

=> [M¹L¹T^-2] = M^aL^b+c T^-b

on comparing both sides we get

a = 1

b+c = 1

-b = -2 => b = 2

Then c = -1 

 Onputting these values in (i),

we get

F = k m¹ v² r^-1

Or

{tex}F = k {mv^2\over r}{/tex}

 This is the required relation for centripetal force.

• The quantities that do not depend on any other physical quantity for their measurement are known as fundamental quantities.These quantities do not take support of other physical quantities for its measurement. There are only 7 physical quantities.
• Afundamental unit is a unit of measurement for a measurable physical property from which every other unit for that quantity can be derived.Fundamental units are also known as base units.

The International System of Units, symbolized SI, is the simplified modern version of the metric system.

• No conversions (only one unit for each quantity)
• No numbers to memorize (derived units are defined without numerical factors)
• No fractions (decimals only)
• No long rows of zeros (prefixes eliminate them) Only 30 individual units (compared to hundreds of traditional units)
• Easy to pronounce and write (short names; simple letter symbols)
• Based on natural standards (size of Earth, water, laws of physics)
• Coherent system (symbols can be manipulated algebraically)
• World standard (even traditional U.S. units are defined by it)

He was actually timing how long between the entry and exit of the darker (umbral) shadow. Which to him appeared to be around 2.6 hours. Next, he compared this with the time it takes the moon to move a distance equal to it’s diameter. In the previous article we determined that this was about 1 hour. So he’s left with two numbers:

• time it takes the moon to travel 1 moon diameter = 1 hour
• time it takes the moon to travel 1 earth diameter  = 2.6 hours

How could we figure out the size of the moon from this?

If the times were equal than it would imply that the moon is the same size as the earth. However the time of a lunar eclipse is much longer, which means the earth must be larger. How much larger? We need to setup a basic proportion:

time #1 / time #2 = moon diameter / earth diameter

1 / 2.6 = moon diameter / earth diameter

This led him to claim that the earth was about 8/3 the diameter of the moon. This is pretty close to the actual difference: the earth is about 3.7 times bigger than the moon. The main reason he underestimated was because the umbral shadow is narrower than the earth.

Actual size of the moon

We now know the relative size of the moon compared to earth. We also know the actual size of the earth from a previous calculation. Finally we can determine the approximate size of the moon!

moon diameter = earth Diameter / 3.7

moon diameter = 12742/3.7 = 3444 km

This is very close to the actual diameter of the moon: 3474.8 km

In this method, laser beam is used in place of sound waves to find the distance of moon from earth.

A laser beam is sent towards moon. This transmitted beam from earth is received back on earth after reflection from the moon.

Suppose the time interval between transmission and reception of the beam is t, the velocity of beam in air/vacuum is c and s is the distance of moon from earth, Then

Distance=Velocity x time

2s=c x t or

s= {tex} {ct \over 2}{/tex}

The inertial mass of an object is the basic property of the object that determines its response to a push or a pull or any external force, governed by Newton's second law of motion. 

Thus, the inertia of matter can be used to measure the inertial mass of the object. 

Gravitational mass is the characteristics of the object that determine its response to the gravitational force by which the object is pulled towards the centre of the Earth. 

Significant figures indicate the precision of measurement of a physical quantity which depends on the least count of the measuring instrument. 

The basic ruled are :

1) All the non-zero digits are significant. 

2) All the zeros b/w two non-zero digits are significant, irrespective of the location of the decimal point. 

3) If the number is less than 1, all the zeros on the right of decimal point but to the left of the first non-zero digit are not significant. 

4) The terminal zero in a number, greater than 1, without a decimal point are not significant. 

5) If the number is greater than 1,all the trailing zero in a number with a decimal point are significant. 

The result of all the measurements is an approximate number that contains some uncertainty, called Error in it. 

We have Systematic errors that tend to be in one direction, either positive or negative. Causes of Systematic errors are instrumental errors, imperfection in experimental techniques etc. 

The other type of errors are random errors that may arise due to unpredictable fluctuations and variations in experimental conditions during the conduct of the experiment. 

In view of the tremendous accuracy required in time measurements and to meet the need for an improved and more precise standard of time, atomic clocks have been developed. 

An error may be defined as the difference between the measured value and the actual value. For example, if the two operators use the same device or instrument for finding the errors in measurement, it is not necessary that they may get the similar results. There may be a difference between both measurements. The difference that occurs between both the measurements is referred to as an ERROR.

1 Joule = 10⁷ erg

25 Joule = 25 × 10⁷ erg = 2.5 × 10⁸ erg

a---[L]

b----[LT^-1]

C--[LT^-2]

• Astronomical Unit(a.u): A unit of measurement equal to 149.6 million kilometres, the mean distance from the centre of the earth to the centre of the sun.
• Angstrom Å ; A unit of length equal to one hundred-millionth of a centimetre, 10⁻¹⁰ metre, used mainly to express wavelengths and interatomic distances.

The efficient cesium atomic clocks are very accurate enough and such tremendous accuracy by the cesium atomic clocks are able to provide portable time measurement standard. 

[a]=M⁰L¹T^-1

^{[b]=M​​​​​0L1T-2}

[a]=M⁰L¹T^-1

Tension is as same as force. So dimensional formula for tension and force is MLT-2. 

The surface tension of water is 72 dync/cm. the value in S.I. unit is

72 × 10^-3 N/m

ML-1T-2

Rate of flow is defined as the quantity of a fluid flowing per second through a section of a pipe or a channel.

Rate of Flow ={tex}Volume \over time {/tex}= m³/s

Putting these values in above equation we get,

Dimensional Formula of Rate of Flow= M⁰L³T^-1