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The above equation consisting of basic constants of atomic physics and the gravitational constant G has the dimension of time.
e = charge of Electron =
= absolute permittivity =
= Mass of the Proton =
= Mass of the Electron =
c = Speed of light in vacuum =
G = Universal Gravitational constant =
Considering T as the age of the universe and putting the values of the constants, we...

From examples 2.3 and 2.4, we have,
Diameter of the Earth =
Distance between the Moon and the Earth, =
Distance between the Sun and the Earth, =
Diameter of the Sun, =
Let, Diameter of the Moon be
Now, During Solar eclipse, the angle subtended by Sun's diameter on Earth = angle subtended by moon's diameter
Therefore, the diameter of the moon =

Let the distance of a quasar from Earth be D km.
We know , Speed of light =
And, time taken by light to reach us , t = 3.0 billion years =
D =Speed of Light x t
=
=

Given,
77.0 s is the total time between the generation of a probe wave and the reception of its echo after reflection.
Time taken by sound to reach the enemy submarine = Half of the total time =
The distance of enemy ship = Speed of sound x Time taken to reach the submarine
=

2.56 s is the total time taken by the LASER to reach Moon and again back Earth.
Time taken by LASER to reach Moon =
We know, Speed of light
The radius of the lunar orbit around the Earth = distance between Earth and Moon =
= Speed of light x Time taken by laser one-way =

The equation for the radius of the nucleus is given by,
The volume of the nucleus using the above relation,
We know,
Mass = Mass number× Mass of single Nucleus
= (given)
Nuclear mass Density = Mass of nucleus/ Volume of nucleus =
The derived density formula contains only one variable, and is independent of mass number A. Since is constant, hence nuclear mass density is nearly constant...

Estimate the average mass density of a sodium atom assuming its size to be about 2.5 Å. (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the mass density of sodium in its crystalline phase: Are the two densities of the same order of magnitude? If so, why? Given, Diameter of sodium = 2.5
Radius, r =
Now, Volume occupied by each atom
We know, One...

In terms of seconds, 100 years =
Given, Difference between the two clocks after 100 years = 0.02 s
In 1 s, the time difference
Accuracy in measuring a time interval of 1 s =
Accuracy of 1 part in

Given,
Mass of the Sun, m =
The radius of the Sun, r =
Volume V =
=
Density = Mass/Volume
=
Therefore, the density of the sun is in the range of solids and liquids and not gases. This high density arises due to inward gravitational attraction on outer layers due to inner layers of the Sun. (Imagine layers and layers of gases stacking up like a pile!)

(a) Height of water column during monsoon is recorded as 215 cm.
H = 215 cm = 2.15 m
Area of the country,
Volume of water column, V = AH
V =
Mass of the rain-bearing clouds over India during the Monsoon, m = Volume x Density
m = = (Density of water = 103 kg m-3 )
b) Consider a large solid cube of known density having a density less than water.
Measure the volume of water displaced...

The statement "Precise measurements of physical quantities are a need of science" is indeed true. In Space explorations, very precise measurement of time in microsecond range is needed. In determining the half-life of radioactive material, very precise value of mass of nuclear particles is required. Similarly, in Spectroscopy precise value of the length in Angstroms is required.

Given, Distance of the star from the solar system = 4.29 ly (light years)
1 light year is the distance travelled by light in one year.
(Note: Light year is a measurement of distance and not time!)
(a)
4.29 ly =
We know, 1 parsec =
4.29 ly = = 1.32 parsec
Now,
(b)
& d = ; D =
Also, We know

The derived formula is dimensionally incorrect.
We know, Trigonometric functions are dimensionless.
Hence , [ ] =
and [v] = .
To make it dimensionally correct, we can divide v by (where is the speed of rain)
Thus, L.H.S and R.H.S are both dimensionless and hence dimensionally satisfied.
The new formula is :

The diameter of Earth’s orbit =
The radius of Earth’s orbit, r =
Let the distance parallax angle be = .
Let the distance between earth and star be R.
(Parsec is the distance at which average radius of earth’s orbit
subtends an angle of .)
We have (Analogous to a circle, R here is the radius, r is the arc length and is the angle covered ! )
Hence, 1 parsec .

Our eyes detect angular velocity, not absolute velocity. An object far away makes a lesser angle than an object which is close. That's why the moon (which is so far away!) does not seem to move at all angularly and thus seems to follow you while driving.
In other words, while in a moving train, or for that matter in any moving vehicle, a nearby object moves in the opposite direction while the...

Radius of hydrogen atom = 0.5 = 0.5 x 10-10 m (Size here refers to Diameter!)
Volume occupied by the hydrogen atom=
=
=
1 mole of hydrogen contains 6.023 x 1023 hydrogen atoms.
Volume of 1 mole of hydrogen atom = 6.023 x 1023 x 0.524 x 10-30
= 3.16 x 10-7 m3
The molar volume is times greater than the atomic volume.
Hence, intermolecular separation in gas is much larger than the size of...

Radius of an Hydrogen atom = 0.5 = 0.5 x 10-10 m
Volume =
=
=
1 hydrogen mole contains hydrogen atoms.
The volume of 1 mole of hydrogen atom =
= .

The relation given is
Divide both sides by ; L.H.S becomes which is dimensionless.
Hence, R.H.S must be dimensionless too. (After Dividing by !)
can be dimensionless only when
Therefore, the dimensional equation is

Ground rules:
[y] = L ()
[a] = L
[v] =
[t/T] is Dimenionless.
(a) The dimensions on both sides are equal, the formula is dimensionally correct.
(b) [vt] = ()(T) = L ( is not dimensionless)
The formula is dimensionally incorrect
(c) [a/T] = (L)/(T)
It is dimensionally incorrect, as the dimensions on both sides are not equal.
(d) The dimensions on both sides are equal, the...

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