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].

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

First equation of uniform accelerated motion

v = u + at

Second equation of uniform accelerated motion

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

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.

Radioactive decay

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

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

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

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

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

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.

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.

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

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

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.

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,

$$ \Delta v= \pm 1 $$

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 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 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.