## Impulse

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

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.

## Learn some extra:

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

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.

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

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.

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

As we know that,
F = ma
So, 1N = 1 kg × 1m s-2
Or, 1N = 103g × 102cm s-2
Or, 1N = 105 g cm s-2
Or, 1N = 105 dyne
So, the 1N is equal to 105 dyne.

## Linear momentum

Linear momentum of a moving particle is given by,

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

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

## Galileo’s law of inertia

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 acceleration

Angular accceleration is given by,

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

SI unit of angular acceleration is rad s-2. Its dimensional formula is [M0L0T-2].

## Moment of inertia

Moment of inertia I is given by,

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

## Angular momentum

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

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

## Tangential acceleration

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

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.

## Torque

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

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

## Displacement

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

Δx = x2 – x1

Where,
x1 = Initial position of a particle.
x2 = Final position of a particle.

## Displacement vector

The displacement vector of the particle is given by,

Δr = (x2 – x1 )i + (y2 – y1 )j + (z2 – z1 )k

Magnitude of Δ r is given by

| Δ r | = [ (x2 – x1 )2 + (y2 – y1 )2 + (z2 – z1 )2 ]1/2

## Position vector

Position vector is given by,
r = xi + yj + zk

Where,
i = unit vector along x direction.
j = unit vector along y direction.
k = unit vector along z direction.

## Velocity

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 [ M0L 1T-1 ].

## Acceleration

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/second2 and its dimensional formula is [ M0L1T-2 ].

## Equation of motion

v = u + at

s = ut + ½ at2

#### Third equation of uniform accelerated motion

v2 = u2 + 2as

Where
s = Displacement.
v = Final velocity of the particle.
u = Initial velocity of the particle.
a = Acceleration of the particle

## Relative velocity

The velocity of object 2 relaive to object 1 is given by,
v21 = v2 – v1

The velocity of object 1 relaive to object 2 is given by,
v12 = v1 – v2

## Density

Formula of density is given by,
ρ = m/V
Where,
m = Mass.
V = Volume.

## Projectile motion

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.

## Relative density

Formula of relative density ρr is given by,
ρr= ρsw
Where,
ρs = Density of substance.
ρw = Density of water at 4oC.

## Weight

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

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

## Maxwell equation

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

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

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

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

## Laplace equation

Laplace equation is given by,

2 Φ = 0

Where,
Φ = Scalar function.
∇ = Laplacian operator.

## Lorentz transformation

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

## Periodic motion

#### Examples of periodic motion

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

Displacement current is given by,

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

## Postulate of special theory of relativity

### Postulate of special theory of relativity

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

## Zeeman effect

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.

## AC generator

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.

## Bragg’s equation

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.

## Ohm’s law

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.

## Superconductivity

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

## Venturimeter

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.

## Uncertainty principle

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

## Thermal expansion

Thermal expansion can be three types:

#### Linear expansion

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

#### Area expansion

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

#### Volume expansion

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

## Thermal conductivity

$$H=kA \frac{T_C-T_D}{L}$$
Where,
k is a constant, called thermal conductivity. Si unit of thermal conductivity is Wm-1k-1.

## Power

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

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

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

K.E. = 1/2 ( mv2 )

or K.E. = p2/2m

Where,
m = Mass of the moving body.
v = Velocity of that body.
p = Linear momentum of that particle.

## Mechanical energy

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

E = K + V
E = 1/2 (mv2) + 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.

## Electric dipole

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

## Gauss’s law

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.

## Electric field

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

## Coulomb’s law

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 ε0 is the permitivity of the free space.

## Degrees of freedom

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

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

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

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,

$$\Delta v= \pm 1$$

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

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.

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

### Harmonic oscillator

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

## Binomial Theorem

• ( 1+x )n = 1+nx+[n( n-1)/2!] .x2 + [n(n-1)(n-2)/3!].x3 +……
• ( 1+x )-n = 1-nx+[-n( n+1)/2!] .x2 – [n(n+1)(n+2)/3!].x3 +……

If x<<1 then x2,x3,…. is negligible. so:

• (1+x ) -n ≈ 1-nx
• (1-x ) n ≈ 1-nx
• (1-x ) -n ≈ 1+nx

## Exponential Series

• ex = 1 + x/1! + x2 /2! + x3/3! + …..
• e = 1 + 1/1! + 1/2! + 1/3! + ….
• e = 2.7182
• e-x = 1 – x/1! + x2 /2! – x3/3! + …..
• ex + e-x = 2 [ 1 + x2/2! + x4/4 + …..]

## Conversion factor

#### Conversion of length

• 1 centimetre = 10-2 metre
• 1 millimetre = 10-3metre
• 1 micrometre = 10-6metre
• 1 nanometre = 10-9 metre
• 1 angstrom= 10-10 metre
• 1 fermi = 10-15 metre
• 1 kilometre = 103 metre
• 1 austronomical unit = 1AU=1.496 × 1011 metre
• 1 light year = 1 ly = 9.461 ×1015metre
• 1 mile = 1.609 ×103 metre
• 1 yard = 0.9144 metre
• 1 inch = 0.0254 metre

#### Conversion of time

• 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× 107 second
• 1 sec = 10-8second

#### Conversion of mass

• 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 × 1030 kg
• 1 atomic mass unit = 1u = 1.66×10-10

## Operators in quantum mechanics

### Operators in quantum mechanics

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

## What is the formula of gravitational potential energy?

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.

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

#### Law of elliptical orbits

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

#### Law of Area

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

#### Law of periods

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.

## What is the formula of Escape speed?

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.

## What is acceleration due to gravity?

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

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

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

$$g’ = g (1 – \frac{2h}{R_e})$$

Where,
Re is the radius of the earth.
g is the acceleratin due to gravity.

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

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

$$g’ = g (1 – \frac{d}{R_e})$$

Where,
Re is the radius of the earth.
g is the acceleratin due to gravity.

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

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

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

Where,
Re is the radius of the earth.
g is the acceleratin due to gravity.
λ is the lattitude of earth

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

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/sec2.

### Gravitational potential energy

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

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.

### Kepler’s law of planetary motion

#### Law of elliptical orbits

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

#### Law of Area

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

#### Law of periods

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.

## What is Schrodinger equation?

Suppose a particle of mass m, constrained to move along the x-direction in the region of the potential V. Then Schrodinger equation for this particle is,

$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V\psi$$

Where,
$$i$$ is the square root of -1 and $$\hbar = \frac{h}{2\pi}$$.
$$\psi$$ is the wave function of the particle.
V is the specified potential.

## What is the dimensional formula ?

The dimensional formula of the physical quantity is an expression which shows how and which of the base quantities represent the dimensions of a physical quantity.

For example [ M0L2T0 ] is the dimensional formula of velocity. similarly, [ M0L3T0 ] is the dimensional formula of volume.

### Dimensional formula of all important physical quantities.

Dimensional formula of major physical quantities e.g. area, volume, velocity, acceleration, force, momentum, impulse, work, energy, power, surface tension etc. are following:

Physical quantity Dimensional formula
Acceleration M0L1T-2
Acceleration due to gravity M0L1T-2
Angle M0L0T0
Angular acceleration M0L0T-2
Angular displacement M0L0T0
Angular frequency M0L0T-1
Angular impulse M1L2T-1
Angular momentum M1L2T-1
Angular velocity M0L0T-1
Area M0L2T0
Binding energy of nucleus M1L2T-2
Boltzmann constant M1L2T-2K-1
Bulk modulus M-1L0T-2
Capacity M-1L-2T4A2
Capacitative reactance M1L2T-3A-2
Centripetal acceleration M0L1T-2
Charge M0L0T1A1
Coefficient of elasticity M1L-1T-2
Coefficient of mutual inductance M1L2T-2A-2
Coefficient of self inductance M1L2T-2A-2
Conductivity M-1L-2T3A2
Critical velocity M0L1T-1
Current density M0L-2T0A1
Decay constant M0L0T-1
Density M1L-3T0
Dielectric constant M0L0T0
Efficiency M0L0T0
Electrical resistivity M1L3T-3A-2
Electric current M0L0T0A1
Electric dipole moment M0L1T1A1
Electric field M1L1T-3A-1
Electric intensity M1L1T-3A-1
Electric potential M1L2T-3A-1
Electric permittivity of free space M-1L-3T4A2
Energy M1L2T-2
Energy density M1L-1T-2
Entropy M1L2T-2K-1
Escape speed M0L1T-1
Frequency M0L0T-1
Force M1L1T-2
Force constant M1L0T-2
Gas constant M1L2T-2K-1mol-1
Gravitational constant M-1L3T-2
Heat M1L2T-2
Heat capacity M1L2T-2K-1
Hubble constant M0L0T-1
Illuminance M1L0T-3
Illuminating power of source M1L2T-3
Impulse M1L1T-1
Inductance M1L2T-2A-2
Inductive reactance M1L2T-3A-3
Intensity of illumination M1L0T-3
Intensity of wave M1L0T-3
Internal energy M1L2T-2
Kinetic energy M1L2T-2
Kinematic viscosity M0L2T-1
Latent heat M0L2T-2
Linear mass density M1L-1T0
Luminance M1L0T-3
Luminous efficiency M0L0T0
Luminous flux M1L2T-3
Luminous intensity M1L2T-3
Luminous power M1L2T-3
Magnification M0L0T0
Magnetic dipole moment M0L2T0A1
Magnetic induction M1T-2A-1
Magnetic intensity A1L-1
Magnetic field M1L0T-2A-1
Magnetic field strength M0L-1T0A1
Magnetic flux M1L2T-2A-1
Magnetic permeability of free space M1L1T-2A-2
Mass defect M1L0T0
Mechanical equivalent of heat M0L0T0
Moment of force M1L2T-2
Moment of inertia M1L2T0
Momentum M1L1T-1
Planck constant M1L2T-1
Pole strength M0L1T0A1
Potential energy M1L2T-2
Power M1L2T-3
Power of lens M0L-1T0
Pressure M1L-1T-2
Pressure energy M1L2T-2
Quality factor M0L0T0
Rate of flow M0L3T-1
Refractive index M0L0T0
Relative Luminosity M0L0T0
Resistance M1L2T-3A-2
Reasonant frequency M0L0T-1
Reynolds number M0L0T0
Rydberg constant M0L-1T0
Specific gravity M0L0T0
Specific heat M0L2T-2K-1
Specific resistance or resistivity M1L3T-3A-2
Specific volume M-1L3T0
Speed M0L1T-1
Stefn’s constant M1L0T-3K-4
Surface density of charge M0L-2T1A1
Surface potential M0L2T-2
Temperature M0L0T0A1
Thermal expansion coefficient M0L0K-1
Torque M1L2T-2
Trigonometric ratio M0L0T0
Velocity M0L1T-1
Velocity of light in vaccum M0L1T-1
Viscosity M1L-1T-1
Volume M0L3T0
Wave number M0L-1T0
Wien’s constant M0L1T0K1
Work M1L2T-2

## What is the Universal law of gravitation?

Force of attraction between two masses $$m_1$$ and $$m_2$$ is given by,

$$F = \frac{m_1 m_2}{r^2}$$

Where,
r is the distance between two masses $$m_1$$ and $$m_2$$.
G is a constant, called the Universal gravitational constant.

That is called Universal law of gravitation.

## What does elasticity mean?

The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed is known as elasticity.

#### Hooke’s law

Within elastic limit, the strain is proportional to stress. That is Hooke’s law.

Stress ∝ Strain
Stress = k × Strain

Where k is constant, called the Modulus of the elasticity.

Elastic energy per unit volume is given by,
ε = ½ × Y × σ2
Where,
Y is young’s modulus of elasticity.
σ is the strain.

#### Modulus of elasticity

According to Hooke’s law

Stress ∝ Strain

or Stress = k × Strain

$$k = \frac{Stress}{Strain}$$

Where k is a constant, called the modulus of elasticity or coefficient of elasticity.

### Types of Modulus of elasticity

#### 1. Young’s modulus of elasticity or Young’s modulus

Young’s modulus of elasticity or Young’s modulus is given by,

$$Y = \frac{\sigma}{\epsilon}$$

Where $$\sigma$$ is normal stress and $$\epsilon$$ is longitudinal strain.

We know that

$$\sigma = \frac{F}{A}$$ and $$\epsilon =\frac{\Delta L}{L}$$

So $$Y = \frac{F L}{A \Delta L}$$

#### 2. Bulk modulus

Bulk modulus is given by,

$$B = \frac{\sigma_v}{\epsilon_v}$$

Where $$\sigma_v$$ is normal stress and $$\epsilon_v$$ is volumetric strain.

We know that

$$\sigma_v = \frac{F}{A}$$ and $$\epsilon_v = \frac{\Delta V}{V}$$

So $$B = \frac{F V}{A \Delta V}$$

#### Modulus of rigidity

Modulus of rigidity is given by

$$G = \frac{\sigma_s}{\theta}$$

Where $$\sigma_s$$ is tangetial stress and $$\theta$$ is shearing strain.

#### Poisson’s ratio

Poisson’s ratio is given by,
$$p.r. =-\frac{d/D}{l/L}$$
where, l/L is the longitudinal strain and d/D is the lateral strain.

## How many fundamental and supplementary units in SI?

In SI, there are seven base ( fundamental ) units and two supplementary units.

#### Base unit

Physical quantity Unit Symbol
Mass Kilogram kg
Length Metre m
Time Second s
Temperature Kelvin K
Electric current Ampere A
Luminous intensity Candela cd
Quantity of matter Mole mol

#### Supplementary units

Physical quantity Unit Symbol

## What is a semiconductor?

### What is a semiconductor?

Those material, which have resistivity or conductivity intermediate to metals or insulator, called semiconductor. The band gap of semiconductor is less than 3 eV. The band gap of Ge and Si respectively 1.1 eV and 0.7 eV.
Si, Ge, GaAs, CdTe etc are examples of semiconductors.
The band gap of Ge and Si respectively 1.1 eV and 0.7 eV.

Si, Ge and C are elemental semiconductor. Examples of compound semiconductors are following:
Inorganic: Cds, GaAs, Cdse, InP etc.
Organic: Anthracene, doped pthalocyanines etc.
Organic Polymers: Polypyrolle, Polyaniline, Polythiophene etc.

An intrinsic semiconductor is one which is made of the semiconductor material in its totally pure form and with no impurities or lattice defects.

If we added small amount ( parts per million, ppm ) of suitable impurities in pure semiconductor , Then this process is called dopping and suitable impurity is called dopent. Dopped intrinsic or pure semiconductor is called extrinsic semiconductor.

There are two types of extrinsic semiconductors-

(a) n-type semiconductor

The pentavalent impurities eg. Phosphorus, Arsenic, Antimony, Bismith etc are referred to as donor impurities. If pure semiconductor is dopped with donor impurities then this type of semiconductor is called n-type semiconductor.

(b) p-type semiconductor

Trivalent impurities eg. Boron, Aluminium, Indium and galium are referred as accepter impurities. If pure semiconductor is dopped with acceptor impurities then this type of semiconductor is called p-type semiconductor.

## Logrithm formula

### Formulae of Logarithm

• loga mn = loga m + loga n
• loga m/n = loga m – loga n
• loga mn = n loga m
• loga m= logb m × loga b
• loge m = 2.3026 log10 m
• log10 m = 0.4343 loge m

### Logrithmic Series

• loge ( 1+x ) = x – x2/2 + x3/3 – x4/4 +……
• loge ( 1-x ) = -[x + x2/2 + x3/3 + x4/4 +……]
• loge ( 1+x ) / ( 1-x ) = 2 [x + x3/3 + x5/5 +……]

## Factors

• ( a+b )2 = a2 + b2 + 2ab
• ( a-b )2 = a2 + b2 – 2ab
• ( a2 – b2) = ( a+b ) ( a-b )
• ( a2 + b2 ) = ( a+b )2– 2ab
• ( a+b )3 = a3 + b3 + 3ab( a+b )
• ( a-b )3 = a3 – b3 – 3ab( a-b )
• ( a+b+c)2 = a2 + b2 + c2 +2(ab + bc + ac )
• a3 + b3 + c3 – 3 abc = ( a+b+c ) ( a2 + b2 + c2 – ab – bc – ac )
• ( a+b )4 = a4 + b4 + 2ab ( 2a2 + 3ab + 2b2)
• ( a-b )4 = a4 + b4 – 2ab ( 2a2 + 3ab – 2b2 )

## Differentiation formula

### Fomulae of Differentiation

• ddx (c) = 0   Where c is constant.
• ddx (cx) = c   Where c is constant.
• dudt = dudxdxdt
• ddx (u+v) = dudx + dvdx
• ddx (uv) = u dvdx + v dudx
• ddx ( xn ) = n xn-1  Where n is real number.
• ddx un = n un-1  dudx   Where u is function of x.
• ddx sin x = cos x
• ddx cos x = -sin x
• ddx tan x = sec2 x
• ddx cot x = – cosec2 x
• ddx sec x = tan x sec x
• ddx cosec x = – cot x cosec x
• ddx loge x = 1x
• ddx loge u = 1u   dudx
• ddx ( ex ) = ex
• ddx ( eax ) = a eax
• ddx sin ax = a cos ax
• ddx sin (ax+b) = a cos (ax+b)
• ddx cos ax = – a sin ax
• ddx cos (ax+b) = – a sin (ax+b)

## How many elements are there in the periodic table?

There are 118 chemical elements in the periodic table.
List of all chemical elements is given below.

Elements of periodic table
Atomic number Element Symbol Element Name
1 H Hydrogen
2 He Helium
3 Li Lithium
4 Be Beryllium
5 B Boron
6 C Carbon
7 N Nitrogen
8 O Oxygen
9 F Fluorine
10 Ne Neon
11 Na Sodium
12 Mg Magnesium
13 Al Aluminium
14 Si Silicon
15 P Phosphorus
16 S Sulfur
17 Cl Chlorine
18 Ar Argon
19 K Potassium
20 Ca Calcium
21 Sc Scandium
22 Ti Titanium
24 Cr Chromium
25 Mn Manganese
26 Fe Iron
27 Co Cobalt
28 Ni Nickel
29 Cu Copper
30 Zn Zinc
31 Ga Gallium
32 Ge Germanium
33 As Arsenic
34 Se Selenium
35 Br Bromine
36 Kr Krypton
37 Rb Rubidium
38 Sr Strontium
39 Y Yttrium
40 Zr Zirconium
41 Nb Niobium
42 Mo Molybdenum
43 Tc Technetium
44 Ru Ruthenium
45 Rh Rhodium
47 Ag Silver
49 In Indium
50 Sn Tin
51 Sb Antimony
52 Te Tellurium
53 I Iodine
54 Xe Xenon
55 Cs Caesium
56 Ba Barium
57 La Lanthanum
58 Ce Cerium
59 Pr Praseodymium
60 Nd Neodymium
61 Pm Promethium
62 Sm Samarium
63 Eu Europium
65 Tb Terbium
66 Dy Dysprosium
67 Ho Holmium
68 Er Erbium
69 Tm Thulium
70 Yb Ytterbium
71 Lu Lutetium
72 Hf Hafnium
73 Ta Tantalum
74 W Tungsten
75 Re Rhenium
76 Os Osmium
77 Ir Iridium
78 Pt Platinum
79 Au Gold
80 Hg Mercury
81 Tl Thallium
83 Bi Bismuth
84 Po Polonium
85 At Astatine
87 Fr Francium
89 Ac Actinium
90 Th Thorium
91 Pa Protactinium
92 U Uranium
93 Np Neptunium
94 Pu Plutonium
95 Am Americium
96 Cm Curium
97 Bk Berkelium
98 Cf Californium
99 Es Einsteinium
100 Fm Fermium
101 Md Mendelevium
102 No Nobelium
103 Lr Lawrencium
104 Rf Rutherfordium
105 Db Dubnium
106 Sg Seaborgium
107 Bh Bohrium
108 Hs Hassium
109 Mt Meitnerium
111 Rg Roentgenium
112 Cn Copernicium
113 Nh Nihonium
114 Fl Flerovium
115 Mc Moscovium
116 Lv Livermorium
117 Ts Tennessine
118 Og Oganesson

### s-block elements

s-block elements list:

hydrogen (H)
lithium (Li)
helium (He)
sodium (Na)
beryllium (Be)
potassium (K)
magnesium (Mg)
rubidium (Rb)
calcium (Ca)
cesium (Cs)
strontium (Sr)
francium (Fr)

Class 1
Atomic number Symbol Electronic configuration
3 Li [He]2s1
11 Na [Ne]3s1
19 K [Ar]4s1
37 Rb [Kr]5s1
55 Cs [Xe]6s1
87 Fr [Rn]7s1
Class 2
Atomic number Symbol Electronic configuration
4 Be [He]2s2
12 Mg [Ne]3s3
20 Ca [Ar]4s
38 Sr [Kr]5s2
56 Ba [Xe]6s2
88 Ra [Rn]7s2

### p-block elements with atomic number and symbol

Group 13
Atomic number Symbol Name
5 B Boron
13 Al Aluminium
31 Ga Gallium
49 In Indium
81 Tl Thallium
Group 14
Atomic number Symbol Name
6 C Carbon
14 Si Silicon
32 Ge Germanium
50 Sn Tin
Group 15
Atomic number Symbol Name
7 N Nitrogen
15 P Phosphorus
33 As Arsenic
51 Sb Antimony
83 Bi Bismuth
Group 16
Atomic number Symbol Name
8 O Oxygen
16 S Sulfur
34 Se Selenium
52 Te Tellurium
84 Po Polonium
Group 17
Atomic number Symbol Name
9 F Fluorine
17 Cl Chlorine
35 Br Bromine
53 I Iodine
85 At Astatine
Group 18
Atomic number Symbol Name
2 He Helium
10 Ne Neon
18 Ar Argon
36 Kr Krypton
54 Xe Xenon

### Electronic configuration of p-block elements

Group 13
Atomic number Symbol Electronic configuration
5 B [He]2s22p1
13 Al [Ne]3s23p1
31 Ga [Ar]3d104s24p1
49 In [Kr]4d105s25p1
81 Tl [Xe]4f145d106s26p
Group 14
Atomic number Symbol Electronic configuration
6 C [He]2s22p2
14 Si [Ne]3s23p2
32 Ge [Ar]3d104s24p2
50 Sn [Kr]4d105s25p3
82 Pb [Xe]4f145d106s2 6p2
Group 15
Atomic number Symbol Electronic configuration
7 N [He]2s22p3
15 P [Ne]3s23p3
33 As [Ar]3d104s24p3
51 Sb [Kr]4d105s25p3
83 Bi [Xe]4f145d106s2 6p3
Group 16
Atomic number Symbol Electronic configuration
8 O [He]2s22p4
16 S [Ne]3s23p4
34 Se [Ar]3d104s24p4
52 Te [Kr]4d105s25p4
84 Po [Xe]4f145d106s2 6p4
Group 17
Atomic number Symbol Electronic configuration
9 F [He]2s22p5
17 Cl [Ne]3s23p5
35 Br [Ar]3d104s24p5
53 I [Kr]4d105s25p5
85 At [Xe]4f145d106s2 6p5
Group 18
Atomic number Symbol Electronic configuration
2 He 1s2
10 Ne [He]2s22p6
18 Ar [Ne]3s23p6
36 Kr [Ar]3d104s24p6
54 Xe [Kr]4d105s25p6
86 Rn [Xe]4f145d106s2 6p6
Group 13
Atomic number Symbol Name
5 B Boron
13 Al Aluminium
31 Ga Gallium
49 In Indium
81 Tl Thallium
Group 14
Atomic number Symbol Name
6 C Carbon
14 Si Silicon
32 Ge Germanium
50 Sn Tin
Group 15
Atomic number Symbol Name
7 N Nitrogen
15 P Phosphorus
33 As Arsenic
51 Sb Antimony
83 Bi Bismuth
Group 16
Atomic number Symbol Name
8 O Oxygen
16 S Sulfur
34 Se Selenium
52 Te Tellurium
84 Po Polonium
Group 17
Atomic number Symbol Name
9 F Fluorine
17 Cl Chlorine
35 Br Bromine
53 I Iodine
85 At Astatine
Group 18
Atomic number Symbol Name
2 He Helium
10 Ne Neon
18 Ar Argon
36 Kr Krypton
54 Xe Xenon

### Electronic configuration of p-block elements

Group 13
Atomic number Symbol Electronic configuration
5 B [He]2s22p1
13 Al [Ne]3s23p1
31 Ga [Ar]3d104s24p1
49 In [Kr]4d105s25p1
81 Tl [Xe]4f145d106s26p
Group 14
Atomic number Symbol Electronic configuration
6 C [He]2s22p2
14 Si [Ne]3s23p2
32 Ge [Ar]3d104s24p2
50 Sn [Kr]4d105s25p3
82 Pb [Xe]4f145d106s2 6p2
Group 15
Atomic number Symbol Electronic configuration
7 N [He]2s22p3
15 P [Ne]3s23p3
33 As [Ar]3d104s24p3
51 Sb [Kr]4d105s25p3
83 Bi [Xe]4f145d106s2 6p3
Group 16
Atomic number Symbol Electronic configuration
8 O [He]2s22p4
16 S [Ne]3s23p4
34 Se [Ar]3d104s24p4
52 Te [Kr]4d105s25p4
84 Po [Xe]4f145d106s2 6p4
Group 17
Atomic number Symbol Electronic configuration
9 F [He]2s22p5
17 Cl [Ne]3s23p5
35 Br [Ar]3d104s24p5
53 I [Kr]4d105s25p5
85 At [Xe]4f145d106s2 6p5
Group 18
Atomic number Symbol Electronic configuration
2 He 1s2
10 Ne [He]2s22p6
18 Ar [Ne]3s23p6
36 Kr [Ar]3d104s24p6
54 Xe [Kr]4d105s25p6
86 Rn [Xe]4f145d106s2 6p6

### d-block elements

List of d-block elements:

Scandium

Titanium

Chromium

Manganese

Iron

Cobalt

Nickel

Copper

Zinc

Yttrium

Zirconium

Platinum

Gold

Mercury

Rutherfordium

Dubnium

Seaborgium

Bohrium

Hassium

Meitnerium

Ununbium

Niobium

Iridium

Molybdenum

Technetium

Ruthenium

Rhodium

Silver

Hafnium

Tantalum

Tungsten

Rhenium

Osmium

### f-block elements

List of f-block elements:

Scandium
Titanium
Chromium
Manganese
Iron
Cobalt
Nickel
Copper
Zinc
Yttrium
Zirconium
Niobium
Molybdenum
Technetium
Ruthenium
Rhodium
Silver
Lanthanum
Cerium
Praseodymium
Neodymium
Promethium
Samarium
Europium
Terbium
Dysprosium
Holmium
Erbium
Thulium
Ytterbium
Lutetium
Halfnium
Tantalum
Tungsten
Rhenium
Osmium
Iridium
Platinum
Gold
Mercury
Actinium
Thorium
Protactinium
Uranium
Neptunium
Plutonium
Americium
Curium
Berkelium
Californium
Einsteinium
Fermium
Mendelevium
Nobelium
Lawrencium
Rutherfordium
Dubnium
Seaborgium
Bohrium
Hassium
Meitnerium
Roentgenium
Copernicium

## Integration formula

### Integration formula

• ∫ xn dx = xn+1n+1   where ( n≠ -1 )
• ∫ dx = x
• ∫ c xn dx = c xn+1n+1   where ( n≠ -1 )
• 1x dx = ln x
• ∫ sin x dx = – cos x
• ∫ cos x dx = sin x
• ∫ ex dx = ex
• ∫ ( u+v ) dx = ∫ u dx + ∫ v dx
• ∫ sec2 x dx = tan x
• ∫ cosec2 x dx = – cot x
• ∫ sec x tan x dx = sec x
• ∫ cosec x cot x dx = – cosec x
• ∫ eax dx = 1a eax
• ∫ sin ax dx = – 1a cos ax
• ∫ sin (ax+b) dx = – 1a cos (ax+b)
• ∫ cos ax dx = 1a sin ax
• ∫ cos (ax+b) dx = 1a sin (ax+b)

## What is the energy?

The capacity to do work is called energy.
Energy is a scalar quantity. The dimensional formula of the energy are the same as the dimensional formula of work i.e. [ M1L2T-2 ].
SI unit of energy is joule and CGS unit of energy is erg.

### Characteristics of energy

• The entire matter possesss energy.
• Energy can neither be created nor be destroyed. The quantity of energy in the universe is constant
• Energy can be transformed from one form to the other.

## Trigonometry formula

### Basic trigonometry formula

• Perp./Hyp. = sin θ
• Base/Hyp. = cos θ
• Perp./Base = tan θ
• Base/Perp. = cot θ
• Hyp./Base = sec θ
• Hyp./Perp. = cosec θ

### Baudhayana formula

( Hyp. )2 = ( Base )2 + ( Perp.)2

### Some common trigonometric Formulas

• sin(A+B) = sin A cos B + cos A sin B
• sin(A-B) = sin A cos B – cos A sin B
• cos(A+B) = cos A cos B – sin A sin B
• cos(A-B) = cos A cos B + sin A sin B
• sin 2A = 2 sin A cos B
• cos 2A = cos2 A – sin2 A
• cos 2A = 1 – 2 sin2 A
• sin(A+B) + sin(A-B) = 2 sin A cos B
• sin(A+B) – sin(A-B) = 2 cos A sin B
• cos(A+B) + cos(A-B) = 2 cos A cos B
• cos(A+B) – cos(A-B) = -2 sin A sin B

### Trigonometric Ratios

#### Trigonometry ratio table:

Table of trigonometrical ratios of some standard angels:

Angle sin θ cos θ tan θ
00 0 1 0
300 $$\frac{1}{2}$$ $$\frac{\sqrt{3}}{2}$$ $$\frac{1}{\sqrt{3}}$$
450 $$\frac{1}{\sqrt{2}}$$ $$\frac{1}{\sqrt{2}}$$ 1
600 $$\frac{\sqrt{3}}{2}$$ $$\frac{1}{2}$$ $$\sqrt{3}$$
900 1 0 $$\infty$$
1200 $$\frac{\sqrt{3}}{2}$$ $$-\frac{1}{2}$$ $$-\sqrt{3}$$
1350 $$\frac{1}{\sqrt{2}}$$ $$-\frac{1}{\sqrt{2}}$$ -1
1500 $$\frac{1}{2}$$ $$-\frac{\sqrt{3}}{2}$$ $$-\frac{1}{\sqrt{3}}$$
1800 0 -1 0
2700 -1 0 $$-\infty$$
3600 0 1 0
Angle cot θ sec θ cosec θ
00 $$\infty$$ 1 $$\infty$$
300 $$\sqrt{3}$$ $$\frac{2}{\sqrt{3}}$$ 2
450 1 $$\sqrt{2}$$ $$\sqrt{2}$$
600 $$\frac{1}{\sqrt{3}}$$ 2 $$\frac{2}{\sqrt{3}}$$
900 0 $$\infty$$ 1
1200 $$-\frac{1}{\sqrt{3}}$$ -2 $$\frac{2}{\sqrt{3}}$$
1350 -1 $$-\sqrt{2}$$ $$\sqrt{2}$$
1500 $$-\sqrt{3}$$ $$-\frac{2}{\sqrt{3}}$$ 2
1800 $$-\infty$$ -1 $$\infty$$
2700 -1 0 $$\infty$$
3600 $$\infty$$ 1 $$\infty$$

#### Relation between Trigonometric Ratios

• sin θ cosec θ = 1
• cos θ sec θ = 1
• tan θ cot θ = 1
• tan θ = sin θ/cos θ
• cot θ = cos θ/sin θ
• sin2 θ + cos2 θ = 1
• 1 + tan2 θ = sec2 θ
• 1 + cot2 θ = cosec2 θ

#### A. Trigonometric Ratios of acute angles

• Perp./Hyp. = sin θ
• Base/Hyp. = cos θ
• Perp./Base = tan θ
• Base/Perp. = cot θ
• Hyp./Base = sec θ
• Hyp./Perp. = cosec θ

#### B. Trigonometric ratios of allied angles

1. Trigonometric ratios of (-θ) in terms of (θ)

sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
cosec(-θ) = -cosecθ

2. Trigonometric ratios of (900-θ) in terms of (θ)

sin(900-θ) = cosθ
cos(900-θ) = sinθ
tan(900-θ) = cotθ
cot(900-θ) = tanθ
sec(900-θ) = cosecθ
cosec(900-θ) = secθ

3. Trigonometric ratios of (900+θ) in terms of (θ)

sin(900+θ) = cosθ
cos(900+θ) = -sinθ
tan(900+θ) = -cotθ
cot(900+θ) = -tanθ
sec(900+θ) = -cosecθ
cosec(900+θ) = secθ

4. Trigonometric ratios of (1800-θ) in terms of (θ)

sin(1800-θ) = sinθ
cos(1890-θ) = -cosθ
tan(1800-θ) = -tanθ
cot(1800-θ) = -cotθ
sec(1800-θ) = -secθ
cosec(1800-θ) = cosecθ

5. Trigonometric ratios of (1800+θ) in terms of (θ)

sin(1800+θ) = -sinθ
cos(1800+θ) = -cosθ
tan(1800+θ) = tanθ
cot(1800+θ) = cotθ
sec(1800+θ) = -secθ
cosec(1800+θ) = -cosecθ

6. Trigonometric ratios of (900+θ) in terms of (θ)

sin(2700-θ) = -cosθ
cos(2700-θ) = -sinθ
tan(2700-θ) = cotθ
cot(2700-θ) = tanθ
sec(2700-θ) = -cosecθ
cosec(2700θ) = -secθ

7. Trigonometric ratios of (900+θ) in terms of (θ)

sin(2700+θ) = -cosθ
cos(2700+θ) = sinθ
tan(2700+θ) = -cotθ
cot(2700+θ) = -tanθ
sec(2700+θ) = cosecθ
cosec(2700+θ) = -secθ

8. Trigonometric ratios of (3600-θ) in terms of (θ)

sin(3600-θ) = -sinθ
cos(3600-θ) = cosθ
tan(3600-θ) = -tanθ
cot(3600-θ) = -cotθ
sec(3600-θ) = secθ
cosec(3600-θ) = -cosecθ

9. Trigonometric ratios of (3600-θ) in terms of (θ)

sin(3600+θ) = sinθ
cos(3600+θ) = cosθ
tan(3600+θ) = tanθ
cot(3600+θ) = cotθ
sec(3600+θ) = secθ
cosec(3600+θ) = cosecθ

10. Trigonometric ratios of (n×3600±θ) in terms of (θ)

sin(n×3600±θ) = ±sinθ
cos(n×3600±θ) = cosθ
tan(n×3600±θ) = ±tanθ
cot(n×3600±θ) = ±cotθ
sec(n×3600±θ) = secθ
cosec(n×3600±θ) = ±cosecθ

#### C. Trigonometric ratios of compound angels

1. Trigonometric ratios of sum and difference of two angles

• sin(A+B) = sinA cosB + cosA sinB
• cos(A+B) = cosA cosB – sinA sinB
• sin(A-B) = sinA cosB – cosA sinB
• cos(A-B) = cosA cosB + sinA sinB

2. Transformation of product into sums of differences

• 2 sinA cosB = sin(A+B) + sin(A-B)
• 2 cosA sinB = sin(A+B) – sin(A-B)
• 2 cosA cosB = cos(A+B) + cos(A-B)
• 2 sinA sinB = cos(A+B) – cos(A-B)

3. Transformation of sum or difference into product

Suppose A+B=C and A-B=D
or $$A = \frac{C+D}{2}$$ and $$B = \frac{C-D}{2}$$

• $$sinC+sinD = 2 sin \frac{C+D}{2} cos\frac{C-D}{2}$$
• $$sinC-sinD = 2 cos \frac{C+D}{2} sin\frac{C-D}{2}$$
• $$cosC+cosD = 2 cos \frac{C+D}{2} cos\frac{C-D}{2}$$
• $$cosC-cosD = 2 sin\frac{C+D}{2} sin\frac{D-C}{2}$$

4. Trigonometric ratios of sum of more than two angles

• sin(A+B+C) = sinA cosB cos C + cosA sinB cosC + cosA cosB sinC – sinA sinB sinC
• cos(A+B+C) = cosA cosB cosC – sinA sinB cosC – sinA cosB sinC – cosA sinB sinC

#### D. Trigonometric ratios of multiple and sub-multiple angles

Multiple angles: 2A, 3A, 4A ……

Sub-multiple angles : $$\frac{A}{2}, \frac{A}{3}, \frac{A}{4}$$…….

1. Trigonometric ratios of an angle 2A in terms of angle A

• sin2A = 2sinA cosA
• cos2A = 1-2sin2A
• $$\ tan2A = \frac{2tanA}{1-tan^2A}$$

2. Trigonometric ratios of sin2A and cos2A in terms of tanA

• $$\ sin2A = \frac{2tanA}{1+tan^2A}$$
• $$\ cos2A = \frac{1-tan^2A}{1+tan^2A}$$

3. Trigonometric ratios of an angle 3A in terms of angle A

• sin3A = 3 sinA – 4sin3A
• cos3A = 4 cos3A – 3 cosA
• $$\ tan3A = \frac{3 tanA – tan^3A}{1-3 tan^2A}$$

4. Trigonometric ratios of an angle 180

• $$sin18^0 = \frac{-1+\sqrt{5}}{4}$$
• $$cos18^0 = \frac{\sqrt{10+2\sqrt{5}}}{4}$$

5. Trigonometric ratios of an angle 360

• $$cos36^0 = \frac{1+\sqrt{5}}{4}$$
• $$sin36^0 = \frac{\sqrt{10-2\sqrt{5}}}{4}$$

6. Trigonometric ratios of an angle A in terms of angle A/2.

• $$sinA = 2 sin\frac{A}{2}cos\frac{A}{2}$$
• $$cosA = 1- 2 sin^2\frac{A}{2}$$
• $$tanA = \frac{2tan\frac{A}{2}}{1-tan^2\frac{A}{2}}$$
• $$sinA = \frac{2tan\frac{A}{2}}{1+tan^2\frac{A}{2}}$$
• $$cosA = \frac{1-tan^2\frac{A}{2}}{1+tan^2\frac{A}{2}}$$

7. Trigonometric ratios of an angle $$\frac{A}{2}$$ in terms of cosA

• $$sin\frac{A}{2} = \pm \sqrt{\frac{1-cosA}{2}}$$
• $$cos\frac{A}{2} = \pm \sqrt{\frac{1+cosA}{2}}$$
• $$tan\frac{A}{2} = \pm \sqrt{\frac{1-cosA}{1+cosA}}$$

8. Trigonometric ratios of an angle $$\frac{A}{2}$$ in terms of sinA

• $$sin\frac{A}{2} + cos\frac{A}{2} = \pm \sqrt{1+sinA}$$
• $$sin\frac{A}{2} – cos\frac{A}{2} = \pm \sqrt{1-sinA}$$

### Series Expnsion of Trigonometric functions

• sin θ = θ – θ3/3! + θ5/5! – θ7/7! …..
• cos θ = 1 – θ2/2! + θ4/4! – θ6/6! …..
• tan θ = θ + θ3/3 + 2θ5/15 …..

If is θ small

• sin θ ≈ θ
• cos θ ≈ 1
• tan θ ≈ θ

### Average Value

• < sin θ > = < sin nθ > = 0
• < cos θ > = < cos nθ > = 0
• < sin2 θ > = < sin2 nθ > = 1/2
• < cos2 θ > = < cos2 nθ > = 1/2

## What are the laws of motion?

#### Newton’s laws of motion

Newton’s gives three law of motion:

##### 1. Newton’s first law motion

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 Newton’s first law motion.

##### 2. Newton’s second law motion

Time-rate of change of momentum is proportional to the applied external force. This is Newton’s second law motion

##### 3. Newton’s third law motion

To every action there is always equal and opposite reaction. This is Newton’s third law motion.

## What is rigid body?

A system of large number of particles in which the distance between any two particles remains fix throughout the motion is called rigid body.

## What is the ideal gas equation?

The equation of state of an ideal gas is given by,

PV = nRT

Where, n is the number of moles of the gas and R is the gas constant for one mole of the gas.

### The gas law

#### Boyle’s law

If $$\mu$$ and T are constant then idael gas equation becomes, $$PV = Constant$$ or $$P \propto \frac{1}{V}$$ This is the Boyle’s law.

#### Charle’s law

If $$\mu$$ and P are constant then idael gas equation becomes,
$$V \propto T$$ This is the Charle’s law.

#### Gay lussac’s law

If $$\mu$$ and V are constant then idael gas equation becomes,
$$P \propto T$$

This is the Gay lussac’s law.

## What are the different types of waves?

Basically waves can be three types:

1. Mechanical waves
2. Electromagnetic waves
3. Matter waves

### Mechanical waves

Waves which can be produced and propagated only in a material medium are known as mechanical waves.
Water waves, sound waves, waves on string, seismic waves etc. are the example of mechanical waves.
The propagation of mechanical wave depends on the elasticity and inertia of the medium. Thus, these waves are known as elastic waves.

Mechanical waves are two types: Transverse waves and longitudinal waves.

#### Transverse waves

Waves on the water surface, light waves, wave generated on the string etc are the examples of transverse waves.

#### Longitudinal wave

Sound wave propagated in air is the example of longitudinal waves.

### Electromagnetic waves

Those waves which requires no material medium for their production and propagation, this means it propagates in vacuum. Such waves are called electromagnetic waves.
Visible light, ultraviolet light, radio waves, microwaves, X-rays etc are the examples of electromagnetic waves.

### Matter waves

These waves are associated with moving particles of matter like electrons, protons, neutrons, atoms, molecules etc.

## What is an electric charge?

Charge is the properties of matter. According to Benjamin franklin there are two types of charge, (1) Positive charge and (2) Negative charge.
Electric charge is a scalar quantity.
In SI System, the unit of Electric charge is Coulomb.
The dimensional formula of Electric charge is [ M0L0T1A1 ]

## What is scalar and vector quantity?

The physical quantity which have only magnitude but no direction, are called scalar quantity.
Mass, length, time, speed, volume, density, pressure, temperature, work, energy, power, electric current, electric charge, electric potential, electric flux etc are the examples of scalar quantity.

### Scalar or Dot Product

Dot product of two vectors A and B is represented by,
A .B = AB cosθ

Where θ is angle between two vectors A and B.

•   If two vectors A and B are parallel, then θ = 0

A.B = AB
For unit vectors,

î . î = ĵ . ĵ = k.k = 1

•   If two vectors A and B are mutually perpendicular, then θ = 90

A.B = 0

For unit vectors,

î . ĵ = ĵ . k = k.i = 0

•   If two vectors A and B are anti parallel, then θ = 0

A.B = – AB

•   Properties of dot product

1.Dot product of two vectors is commutative.

A . B = B . A

2.Dot product is distributive.

A . ( B + C ) = A . B + A . C

•   Dot product of two vectors A and B in component form
A . B = AxBx + AyBy + AzBz

### Cross Product of two vector

Cross product of two $$\vec{A}$$ and $$\vec{B}$$ is represented by,

$$\vec{A} \times \vec{B} = A B \sin \theta \hat{n}$$

Where $$\hat{n}$$ is the unit vector along the resultant vector.

#### If two vectors $$\vec{A}$$ and $$\vec{B}$$ are parallel,

Then $$\theta = 0^o or 180^o$$
So $$\vec{A} \times \vec{B} = 0$$

For unit vectors

$$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$$

#### If two vectors $$\vec{A}$$ and $$\vec{B}$$ are perpendicular,

Then $$\theta = 90^0$$
So $$\vec{A} \times \vec{B} = AB \hat{n}$$

For unit vectors
$$\hat{i} \times \hat{j} = \hat{k}$$ , $$\hat{j} \times \hat{k} = \hat{i}$$ , $$\hat{k} \times \hat{i} = \hat{j}$$

#### Properties of cross product

1. Cross product of two vectors in not commutative.
$$\vec{A} \times \vec{B} = – \vec{B} \times \vec{A}$$
2. Cross product is distributive.
$$\vec{A} \times ( \vec{B} + \vec{C} ) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$$

#### Cross product of two vectors $$\vec{A}$$ and $$\vec{B}$$ in component form

$$\vec{A} \times \vec{B} = ( A_y B_z – A_z B_y ) \hat{i} + ( A_z B_x – A_x B_z ) \hat{j} + ( A_x B_y – A_y B_x ) \hat{k}$$

## What is the errors of measurement in physics?

### What is the errors of measurement in physics?

Errors of measurement = True value of a quantity – Measured value of a quantity

Suppose, the measured value of quantity be Am and the error in measurement be ΔA. Then the true value of the quantity can be written as
At = Am ± ΔA

### Absolute error, Relative error and Percentage error

#### Absolute error

Suppose a physical quantity be measured n times and the measured values be $$a_1, a_2, a_3 — a_n$$. The arithmetic mean of these values is given by,

$$a_m = \frac{a_1 + a_2 + a_3 + … + a_n }{n}$$

The absolute errors in the individual measurement values are

$$\Delta a_1 = a_m – a_1$$

$$\Delta a_2 = a_m – a_2$$

$$\Delta a_3 = a_m – a_3$$

………….

$$\Delta a_n = a_m – a_n$$

#### Mean absolute error

Mean absolute error $$\Delta a_m$$ of a physical quantity is given by,

$$\Delta a_m = \frac {|\Delta a_1| + |\Delta a_2| + |\Delta a_3| +…+ |\Delta a_n|}{n}$$

#### Relative error

Relative error $$R_e$$ is given by,

$$\ R_e = \frac{\Delta a_m}{a_m}$$

#### Percentage error

Percentage error $$p_e$$ is given by,

$$P_e = \frac{\Delta a_m}{a_m} \times 100 \%$$

### Types of errors

1. Systematic errors
2. Random errors
3. Gross errors

## What is work in physics?

### What is work?

Scalar product of force and displacement is called work.

$$W = \vec{F}.\vec{S}\cos \theta$$

Work is a scalar quantity. The dimensional formula of work is [ M1L2T-2 ]. SI unit of work is joule. CGS unit of work is erg.

Relation between joule and erg is
1 joule = 107 erg.

In terms of rectangular components, work is
W = x Fx + y Fy + z Fz

### Types of work

#### Positive work

If angle between $$\vec{F}$$ and $$\vec{S}$$ lies between 00 and 900, then work done is positive.

#### Negative work

If angle between $$\vec{F}$$ and $$\vec{S}$$ lies between 900 and 1800, then work done is negetive.

#### Negative work

The work done is zero, if

• there is no displacement
• no force is acting on the body
• angle between $$\vec{F}$$ and $$\vec{S}$$ is 900

## What is the position vector of centre of mass of the two particle system?

For two particle system, the position vector of centre of mass of the two particle system is given by,

$$\vec{r} = \frac {m_1 \vec{r_1} + m_2 \vec{r_2}}{m_1+m_2}$$

For two particle system, velocity of centre of mass is given by,

$$\vec{v_c} = \frac {m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1+m_2}$$

Whewe,

$$m_1$$ and $$m_2$$ are the masses of two particles.

$$\vec{r_1}$$ and $$\vec{r_2}$$ are the position vector of the particles $$m_1$$ and $$m_2$$ respectively.

$$\vec{v_1}$$ and $$\vec{v_2}$$ are velocities of the particles $$m_1$$ and $$m_2$$ respectively.

## What is logic gates?

Truth table of AND Gate:

INPUT OUTPUT
A B Y = A.B
0 0 0
0 1 0
1 0 0
1 1 1

Truth table of NOT Gate.

INPUT OUTPUT
A Y
0 1
1 0

Truth table of OR Gate.

INPUT OUTPUT
A B Y = A+B
0 0 0
0 1 1
1 0 1
1 1 1

Truth table of NAND Gate.

INPUT OUTPUT
A B Y
0 0 1
0 1 1
1 0 1
1 1 0

Truth table of NOR Gate.

INPUT OUTPUT
A B Y
0 0 1
0 1 0
1 0 0
1 1 0

Truth table of XOR Gate.

INPUT OUTPUT
A B Y
0 0 0
0 1 1
1 0 1
1 1 0