Impulse is given by,

$$ \vec{I} = \vec{F_a} t$$

Where \( \vec{F_a}\) is the average force acts on the particles and t is the time for which the force acts on the particle.

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# Month: January 2021

## Impulse

## Force

## Learn some extra:

### What is the value of 1N in terms of fundamental units?

### What is the value of 1dyne in terms of cgs units?

### What is the relation between newton and dyne?

## Linear momentum

## Galileo’s law of inertia

## Angular velocity

## Angular displacement

## Angular acceleration

## Moment of inertia

## Angular momentum

### Relation between torque and angular momentum

## Tangential acceleration

## Centripetal acceleration

## Torque

## Displacement

## Displacement vector

## Position vector

## Velocity

## Acceleration

## Equation of motion

#### First equation of uniform accelerated motion

#### Second equation of uniform accelerated motion

#### Third equation of uniform accelerated motion

## Relative velocity

## Density

## Projectile motion

## Relative density

## Weight

## Radioactive decay

## Maxwell equation

## Laplace equation

## Lorentz transformation

## Periodic motion

#### Examples of periodic motion

## Displacement current

## Postulate of special theory of relativity

### Postulate of special theory of relativity

## Zeeman effect

## AC generator

## Bragg’s equation

## Ohm’s law

## Superconductivity

## Venturimeter

## Uncertainty principle

## Thermal expansion

#### Linear expansion

#### Area expansion

#### Volume expansion

## Thermal conductivity

## Power

## Potential energy

## Kinetic energy

## Mechanical energy

## Electric flux

## Electric dipole

## Gauss’s law

## Electric field

## Coulomb’s law

## Degrees of freedom

## Specific heat capacity

## Vibrational-Rotational energy level of a diatomic molecule

## Vibrational energy level of a diatomic molecule

## What is the formula of rotational energy of diatomic molecule?

## What is the equation of motion of harmonic oscillator?

### Harmonic oscillator

## Binomial Theorem

## Exponential Series

## Conversion factor

#### Conversion of length

#### Conversion of time

#### Conversion of mass

## Operators in quantum mechanics

### Operators in quantum mechanics

## What is the formula of gravitational potential energy?

## What are the Kepler’s law of planetary motion?

#### Law of elliptical orbits

#### Law of Area

#### Law of periods

## What is the formula of Escape speed?

## What is acceleration due to gravity?

#### Variation in acceleration due to gravity with height

#### Variation in acceleration due to gravity with depth

#### Variation in acceleration due to gravity with rotation of earth

#### Variation in acceleration due to gravity with shape of earh

### Gravitational potential energy

### Escape speed

### Kepler’s law of planetary motion

#### Law of elliptical orbits

#### Law of Area

#### Law of periods

Impulse is given by,

$$ \vec{I} = \vec{F_a} t$$

Where \( \vec{F_a}\) is the average force acts on the particles and t is the time for which the force acts on the particle.

Force on a particle having mass m is given by,

$$\vec{F} = m\vec{a}$$

Where, \(\vec{a}\) is the acceleration of the particle.

- SI unit of force is Newton ( N ), which is equal to kilogram metre per second ( kg m s
^{-1}).

CGS unit of force is dyne.

As we know that,

F = ma

So, 1N = 1 kg × 1m s^{-2}

Or, 1N = 1 kg m s^{-2}

So, the value of 1N in terms of fundamental units is 1 kg m s^{-2}.

As we know that,

F = ma

So, 1dyne = 1 g × 1cm s^{-2}

Or, 1dyne = 1 g cm s^{-2}

So, the value of 1dyne in terms of cgs units is 1 g cm s^{-2}.

As we know that,

F = ma

So, 1N = 1 kg × 1m s^{-2}

Or, 1N = 10^{3}g × 10^{2}cm s^{-2}

Or, 1N = 10^{5} g cm s^{-2}

Or, 1N = 10^{5} dyne

So, the 1N is equal to 10^{5} dyne.

Linear momentum of a moving particle is given by,

$$ \vec{p} = m\vec{v}$$

Where m is the mass of moving particle with velocity \( \vec{v}\)

A body continues in its state of rest or constant velocity along the same straight line, unless not disturbed by some external force. This is Galileo’s law of inertia.

Angular velocity is given by,

$$\omega = \frac{d\theta}{dt}$$

Angular displacement is given by, $$ \theta = \frac{s}{r} $$

Angular accceleration is given by,

$$\alpha = \frac{d\omega}{dt}$$

SI unit of angular acceleration is rad s^{-2}. Its dimensional formula is [M^{0}L^{0}T^{-2}].

Moment of inertia I is given by,

$$ I = \Sigma m_i r_i^2$$

Angular momentum is given by $$\vec{L} = \vec{r} \times \vec{p}$$ Where \(\vec{p}\) is linear momentum of the particle and \(\vec{r}\) is position vector of the particle.

Relation between torque and angular momentum is given $$\tau = \frac{d\vec{L}}{dt}$$

Tangential acceleration is given by,

$$ a_T = \vec{\alpha}\times\vec{r}$$

Where \(\vec{\alpha}\) is the angular acceleration and \(\vec{r}\) is the position vector.

Centripetal acceleration is given by,

$$a_c = \vec{\omega}\times\vec{v}$$

Where \(\vec{\omega}\) is the angular velocity and \(\vec{v}\) is the linear velocity.

If a force \(\vec{F}\) acts at a point, whose position vector is \(\vec{r}\); the torque due to force

$$\vec{\tau} = \vec{r} \times \vec{F}$$

The displacement Δx of a particle is given by the following formula,

Δx = x_{2} – x_{1}

Where,

x_{1} = Initial position of a particle.

x_{2} = Final position of a particle.

The displacement vector of the particle is given by,

Δ**r** = (x_{2} – x_{1} )i + (y_{2} – y_{1} )j + (z_{2} – z_{1} )k

Magnitude of Δ **r** is given by

| Δ **r** | = [ (x_{2} – x_{1} )^{2} + (y_{2} – y_{1} )^{2} + (z_{2} – z_{1} )^{2} ]^{1/2}

Position vector is given by,

**r** = x**i** + y**j** + z**k**

Where,

**i** = unit vector along x direction.

**j** = unit vector along y direction.

**k** = unit vector along z direction.

Rate of change of displacement with respect to the time is called velocity.

The velocity \(\vec{v}\) of a particle is given by the following formula,

$$ \vec{v} = \frac{\vec{x}}{t}$$

where,

\(\vec{x}\) = displacement of a particle

t = time taken

Velocity is a vector quantity. SI unit of velocity is metre/second and its dimensional formula is [ M^{0}L ^{1}T^{-1} ].

The acceleration \(\vec{a}\) of a particle is given by the following formula,

$$\vec{a} = \frac{\vec{v}}{t}$$

where,

\(\vec{v}\) = velocity of the particle.

t = time taken.

Acceleration is a vector quantity. SI unit of acceleration is metre/second^{2} and its dimensional formula is [ M^{0}L^{1}T^{-2} ].

v = u + at

s = ut + ½ at^{2}

v^{2} = u^{2} + 2as

Where

s = Displacement.

v = Final velocity of the particle.

u = Initial velocity of the particle.

a = Acceleration of the particle

The velocity of object 2 relaive to object 1 is given by,

v_{21} = v_{2} – v_{1}

The velocity of object 1 relaive to object 2 is given by,

v_{12} = v_{1} – v_{2}

Formula of density is given by,

ρ = m/V

Where,

m = Mass.

V = Volume.

When a body is thrown at some initial velocity, it starts moving along the parabolic path under the influence of gravitational force. This motion in a parabolic path is called Projectile Motion and the desired object is called projectile.

Formula of relative density ρ_{r} is given by,

ρ_{r}= ρ_{s}/ρ_{w}

Where,

ρ_{s} = Density of substance.

ρ_{w} = Density of water at 4^{o}C.

The formula of weight is given by,

W = mg

Where,

m = Mass of the object.

g = Acceleration due to gravity.

Alpha decay

$$ _Z^AX \rightarrow {_{Z-2}^{A-4}}Y + _2^4He$$

Beata decay

$$ _Z^AX \rightarrow {_{Z+1}^{A}}Y + e^-$$

Gamma decay

$$ _Z^AX^* \rightarrow {_{Z}^{A}}X + \gamma$$

$$ \nabla . \vec{E} = \frac{\rho}{\epsilon_0}$$

$$ \nabla . \vec{B} = 0$$

$$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

$$ \nabla \times \vec{B} = \mu_0 \vec{J}$$

Laplace equation is given by,

∇^{2} Φ = 0

Where,

Φ = Scalar function.

∇ = Laplacian operator.

$$ x’ = \frac{x-vt}{ \sqrt {1- \frac{v^2}{c^2}}}$$

$$ y’ = y$$

$$ z’ = z$$

$$t’ = \frac{t -\frac{vx}{c^2}}{ \sqrt {1- \frac{v^2}{c^2}}}$$

revolution of earth around the sun, rotation of earth about its polar axis, motion of hands of a clock, motion of moon around the earth etc. are examples of periodic motion.

Displacement current is given by,

$$\vec{J_d} = \frac{\partial \vec{E}}{\partial t}$$

- The laws of physics are the same in all inertial frames.
- The speed of light in vacuum has the same value in all inertial frames.

When a light source giving the line spectrum is placed in an uniform external magnetic field, the spectral line emited by the atoms of the source are split. This splitting of spectral line by a magnetic field is called zeeman effect.

An AC generator is an device that converts mechanical energy into electrical energy in form of alternating current. Working of AC generator is based on the principle of Electromagnetic Induction.

A beam containing x-rays of wavelength λ is incident upon a crystal at an angle θ with a family of brags plane whose spacing is d.

In this case the relation between d, θ and λ is following:

2d sin θ = n λ

Where, n is order of scattered beam.

This relation is called Bragg’s equation.

If current I flow through a conductor wire, then the potential difference between the ends of the conductor is,

V ∝ I

or V = RI

Where, R is constant, called resistance of the conductor wire.

This is called Ohm’s law.

At very low temperature, electrical resistivity of some metal and alloys drops suddenly to zero. This phenomenon is called superconductivity.

The venturimeter is a device to measure the speed of incompressible liquid and rate of flow of liquid through pipes. Working of venturimeter is based on Bernoulli’s theorem.

$$\Delta x.\Delta p_x \geq \frac{\hbar}{2}$$

Thermal expansion can be three types:

$$\alpha_l = \frac{1}{\Delta T} \frac{\Delta l}{l}$$

$$\alpha_a = \frac{1}{\Delta T} \frac{\Delta A}{A}$$

$$\alpha_v = \frac{1}{\Delta T} \frac{\Delta V}{V}$$

$$ H=kA \frac{T_C-T_D}{L}$$

Where,

k is a constant, called thermal conductivity. Si unit of thermal conductivity is Wm^{-1}k^{-1}.

The time rate of doing work is called power.It is scalar quantity.

$$ P = \frac{dw}{dt}$$

SI unit of power is watt ( W ), which is equal to Joule per second ( J s^{-1} ).

Potential energy is given by

V = mgh

Where,

m = mass of the body.

h = height of that body.

g = acceleration due to gravity.

Kinetic energy of a moving body is given by following formula,

K.E. = 1/2 ( mv^{2} )

or K.E. = p^{2}/2m

Where,

m = Mass of the moving body.

v = Velocity of that body.

p = Linear momentum of that particle.

The mechanical energy ( E ) of a body is given by,

E = K + V

E = 1/2 (mv^{2}) + mgh

Where,

K = Kinetic energy of a body.

V = Potential energy.

m = mass of a body.

v = velocity of that body.

g = acceleration due to gravity.

h = height of a body.

$$\Delta \phi = \vec{E}.\Delta S$$

If two equal and opposite charges +q and -q are separated by a distance 2d, then this arrangement is called electric dipole.

Electric flux ( φ ) through a closed surface (S) enclosing the total charge (q) is given by,

$$\phi = \frac{q}{\epsilon_0}$$

That is is the Gauss’s law.

The force per unit charge that would be exerted on a test charge is called electric field.

The force on a test charge Q due to a single point charge q is given by coulomb’s law $$\vec{F} = \frac{1}{4\pi \epsilon_0 } \frac{Qq}{r^2} \hat{r}$$ Where r is the distance between Q and q and ε_{0} is the permitivity of the free space.

Number of degrees of freedom of a system is given by following,

N = 3A – R

Where,

A is the number of particles in the system and R is the number of independent relations among the particles.

For mono atomic gases, the degrees of freedom is three.

For diatomic gases, the degrees of freedom is five.

Linear triatomic molecules has seven degrees of freedom.

A non Linear triatomic molecules has six degrees of freedom.

Specific heat capacity is given by,

$$ s = \frac{S}{m}$$

Where, S is heat capacity and $latex m$ is the mass of substance.

The unit of specific heat capacity is J kg^{-1} k^{-1}

Vibrational-Rotational energy level of a diatomic molecule is given by,

$$\ E_v = ( v + \frac{1}{2} ) \hbar \omega_0 + \frac {\hbar ^2}{2I} J( J+1 )$$

Where, \(v=0,1,2,3….\) is the vibrational quantum number.

\(\omega_0 = \sqrt \frac{k}{\mu}\) , \(\mu\) is the reduced mass of the diatomic molecule and k is the force constant.

I is the Moment of inertia of diatomoc molecule and J is the Rotational quantum number.

Vibrational energy level of a diatomic molecule is given by,

$$ E_v = ( v + \frac{1}{2} ) \hbar \omega_0 $$

Where, v=0,1,2,3…. is the vibrational quantum number, \(\omega_0 = \sqrt \frac{k}{\mu} \) , \( \mu \) is the reduced mass of the diatomic molecule and k is the force constant.

The selection rule for transition between vibrational states is,

$$ \Delta v= \pm 1 $$

Rotational energy of diatomic molecule is given by,

$$ E_J = \frac {\hbar ^2}{2I} J( J+1 )$$

Where,

I = Moment of inertia of diatomoc molecule.

J = Rotational quantum number.

Equation of motion of harmonic oscillator is,

$$ \frac{d^2x}{dt^2}+\frac{k}{m}x=0$$

Frequency of harmonic oscillator is,

$$ \nu = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$$

The energy of a harmonic oscillator is,

$$ E = (n+\frac{1}{2})h\nu$$

- ( 1+x )
^{n}= 1+nx+[n( n-1)/2!] .x^{2}+ [n(n-1)(n-2)/3!].x^{3}+…… - ( 1+x )
^{-n}= 1-nx+[-n( n+1)/2!] .x^{2}– [n(n+1)(n+2)/3!].x^{3}+……

If x<<1
then x^{2},x^{3},…. is negligible. so:

- (1+x )
^{-n}≈ 1-nx - (1-x )
^{n}≈ 1-nx - (1-x )
^{-n}≈ 1+nx

- e
^{x}= 1 + x/1! + x^{2}/2! + x^{3}/3! + ….. - e = 1 + 1/1! + 1/2! + 1/3! + ….
- e = 2.7182
- e
^{-x}= 1 – x/1! + x^{2}/2! – x^{3}/3! + ….. - e
^{x}+ e^{-x}= 2 [ 1 + x^{2}/2! + x^{4}/4 + …..]

- 1 centimetre = 10
^{-2}metre - 1 millimetre = 10
^{-3}metre - 1 micrometre = 10
^{-6}metre - 1 nanometre = 10
^{-9}metre - 1 angstrom= 10
^{-10}metre - 1 fermi = 10
^{-15}metre - 1 kilometre = 10
^{3}metre - 1 austronomical unit = 1AU=1.496 × 10
^{11}metre - 1 light year = 1 ly = 9.461 ×10
^{15}metre - 1 mile = 1.609 ×10
^{3}metre - 1 yard = 0.9144 metre
- 1 inch = 0.0254 metre

- 1 mili second= 10
^{-3}second - 1 micro second = 10
^{-6}second - 1 neno second = 10
^{-9}second - 1 hour = 60 minute = 3600 second
- 1 day = 24 hours =86400 second
- 1 year = 365 day = 3.156× 10
^{7}second - 1 sec = 10
^{-8}second

- 1 gram = 10
^{-3}kg - 1 quintal = 100 kg
- 1 tonne = 1000 kg
- 1 slug = 14.59 kg
- 1 Chandersekhar limit = 1.4 × mass of sun = 2.8 × 10
^{30}kg - 1 atomic mass unit = 1u = 1.66×10
^{-10}

Physical quantity | Operator |
---|---|

Position,x | $$x$$ |

Linear momentum,p | $$-\iota\hbar \frac{\partial}{\partial x}$$ |

Potential energy,V(x) | $$V(x)$$ |

Kinetic energy,KE | $$-\frac{\hbar}{2m} \frac{\partial^2}{\partial x^2}$$ |

Total energy,E | $$\iota\hbar \frac{\partial}{\partial t}$$ |

Total energy (Hamiltonian),H | $$-\frac{\hbar}{2m} \frac{\partial^2}{\partial x^2} + V(x)$$ |

Angular momentum,$$\hat{L_x}$$ | $$-\iota\hbar(y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y})$$ |

Gravitational potential energy of the body of mass m is given by,

$$ U = – \frac{GMm}{r}$$

where,

M is the mass of earth.

r is the distance between M and m and r>R.

Every planets move around the sun in elliptical orbits, the sun being at one of the focus.

The radius vector, drown from the sun to planet, sweeps out equal areas in equal time.

The sqare of the period of the revolution of the planet around the sun is proportional to cube of the semi-major axis of the ellipse.

Escape speed from earth’s surface is given by

$$v_e = \sqrt \frac{2GM}{R}$$

Where,

M is the mass of earth.

R is the radius of the earth.

G is the universal gravitational constant.

If a body is falling freely, under the effect of gravity, then the acceleration in the body is called acceleration due to gravity.

Acceleration due to gravity at a height h above the surface of earth is,

$$g’ = g (1 – \frac{2h}{R_e})$$

Where,

R_{e} is the radius of the earth.

g is the acceleratin due to gravity.

Acceleration due to gravity at a depth d below the surface of earth is,

$$g’ = g (1 – \frac{d}{R_e})$$

Where,

R_{e} is the radius of the earth.

g is the acceleratin due to gravity.

Acceleration due to gravity, when rotation of earh is taken into account is,

$$g’ = g – R_e \omega^2 \cos^2 \lambda$$

Where,

R_{e} is the radius of the earth.

g is the acceleratin due to gravity.

λ is the lattitude of earth

Equatorial radius of the earth is about is 21 km greather than the polar radius. It means value of acceleration due to gravity is increases as we go from equator to the pole.

Acceleration due to gravity on the earth surface is 9.8 m/sec^{2}.

Gravitational potential energy of the body of mass m is given by,

$$ U = – \frac{GMm}{r}$$

where,

M is the mass of earth.

r is the distance between M and m and r>R.

Escape speed from earth’s surface is given by

$$v_e = \sqrt \frac{2GM}{R}$$

Where,

M is the mass of earth.

R is the radius of the earth.

G is the universal gravitational constant.

Every planets move around the sun in elliptical orbits, the sun being at one of the focus.

The radius vector, drown from the sun to planet, sweeps out equal areas in equal time.

The sqare of the period of the revolution of the planet around the sun is proportional to cube of the semi-major axis of the ellipse.