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.
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.
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,
Δr = (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 = xi + yj + zk
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,
$$\vec{J_d} = \frac{\partial \vec{E}}{\partial t}$$
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,
$$ E = (n+\frac{1}{2})h\nu$$
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.
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.
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 [ M^{0}L^{2}T^{0} ] is the dimensional formula of velocity. similarly, [ M^{0}L^{3}T^{0} ] is the dimensional formula of volume.
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 | M^{0}L^{1}T^{-2} |
Acceleration due to gravity | M^{0}L^{1}T^{-2} |
Angle | M^{0}L^{0}T^{0} |
Angular acceleration | M^{0}L^{0}T^{-2} |
Angular displacement | M^{0}L^{0}T^{0} |
Angular frequency | M^{0}L^{0}T^{-1} |
Angular impulse | M^{1}L^{2}T^{-1} |
Angular momentum | M^{1}L^{2}T^{-1} |
Angular velocity | M^{0}L^{0}T^{-1} |
Area | M^{0}L^{2}T^{0} |
Avogadro’s number | M^{0}L^{0}T^{0} |
Binding energy of nucleus | M^{1}L^{2}T^{-2} |
Boltzmann constant | M^{1}L^{2}T^{-2}K^{-1} |
Bulk modulus | M^{-1}L^{0}T^{-2} |
Capacity | M^{-1}L^{-2}T^{4}A^{2} |
Capacitative reactance | M^{1}L^{2}T^{-3}A^{-2} |
Centripetal acceleration | M^{0}L^{1}T^{-2} |
Charge | M^{0}L^{0}T^{1}A^{1} |
Coefficient of elasticity | M^{1}L^{-1}T^{-2} |
Coefficient of mutual inductance | M^{1}L^{2}T^{-2}A^{-2} |
Coefficient of self inductance | M^{1}L^{2}T^{-2}A^{-2} |
Conductivity | M^{-1}L^{-2}T^{3}A^{2} |
Critical velocity | M^{0}L^{1}T^{-1} |
Current density | M^{0}L^{-2}T^{0}A^{1} |
Decay constant | M^{0}L^{0}T^{-1} |
Density | M^{1}L^{-3}T^{0} |
Dielectric constant | M^{0}L^{0}T^{0} |
Efficiency | M^{0}L^{0}T^{0} |
Electrical resistivity | M^{1}L^{3}T^{-3}A^{-2} |
Electric current | M^{0}L^{0}T^{0}A^{1} |
Electric dipole moment | M^{0}L^{1}T^{1}A^{1} |
Electric field | M^{1}L^{1}T^{-3}A^{-1} |
Electric intensity | M^{1}L^{1}T^{-3}A^{-1} |
Electric potential | M^{1}L^{2}T^{-3}A^{-1} |
Electric permittivity of free space | M^{-1}L^{-3}T^{4}A^{2} |
Energy | M^{1}L^{2}T^{-2} |
Energy density | M^{1}L^{-1}T^{-2} |
Entropy | M^{1}L^{2}T^{-2}K^{-1} |
Escape speed | M^{0}L^{1}T^{-1} |
Faraday constant | M^{0}L^{0}T^{1}A^{1}mol^{-1} |
Frequency | M^{0}L^{0}T^{-1} |
Force | M^{1}L^{1}T^{-2} |
Force constant | M^{1}L^{0}T^{-2} |
Gas constant | M^{1}L^{2}T^{-2}K^{-1}mol^{-1} |
Gravitational constant | M^{-1}L^{3}T^{-2} |
Heat | M^{1}L^{2}T^{-2} |
Heat capacity | M^{1}L^{2}T^{-2}K^{-1} |
Hubble constant | M^{0}L^{0}T^{-1} |
Illuminance | M^{1}L^{0}T^{-3} |
Illuminating power of source | M^{1}L^{2}T^{-3} |
Impulse | M^{1}L^{1}T^{-1} |
Inductance | M^{1}L^{2}T^{-2}A^{-2} |
Inductive reactance | M^{1}L^{2}T^{-3}A^{-3} |
Intensity of illumination | M^{1}L^{0}T^{-3} |
Intensity of wave | M^{1}L^{0}T^{-3} |
Internal energy | M^{1}L^{2}T^{-2} |
Kinetic energy | M^{1}L^{2}T^{-2} |
Kinematic viscosity | M^{0}L^{2}T^{-1} |
Latent heat | M^{0}L^{2}T^{-2} |
Linear mass density | M^{1}L^{-1}T^{0} |
Luminance | M^{1}L^{0}T^{-3} |
Luminosity of radiant flux | M^{1}L^{2}T^{-3} |
Luminous efficiency | M^{0}L^{0}T^{0} |
Luminous flux | M^{1}L^{2}T^{-3} |
Luminous intensity | M^{1}L^{2}T^{-3} |
Luminous power | M^{1}L^{2}T^{-3} |
Magnification | M^{0}L^{0}T^{0} |
Magnetic dipole moment | M^{0}L^{2}T^{0}A^{1} |
Magnetic induction | M^{1}T^{-2}A^{-1} |
Magnetic intensity | A^{1}L^{-1} |
Magnetic field | M^{1}L^{0}T^{-2}A^{-1} |
Magnetic field strength | M^{0}L^{-1}T^{0}A^{1} |
Magnetic flux | M^{1}L^{2}T^{-2}A^{-1} |
Magnetic permeability of free space | M^{1}L^{1}T^{-2}A^{-2} |
Mass defect | M^{1}L^{0}T^{0} |
Mechanical equivalent of heat | M^{0}L^{0}T^{0} |
Moment of force | M^{1}L^{2}T^{-2} |
Moment of inertia | M^{1}L^{2}T^{0} |
Momentum | M^{1}L^{1}T^{-1} |
Planck constant | M^{1}L^{2}T^{-1} |
Pole strength | M^{0}L^{1}T^{0}A^{1} |
Potential energy | M^{1}L^{2}T^{-2} |
Power | M^{1}L^{2}T^{-3} |
Power of lens | M^{0}L^{-1}T^{0} |
Pressure | M^{1}L^{-1}T^{-2} |
Pressure energy | M^{1}L^{2}T^{-2} |
Pressure gradient | M^{1}L^{-2}T^{-2} |
Quality factor | M^{0}L^{0}T^{0} |
Radius of gyration | M^{0}L^{1}T^{0} |
Radiant flux | M^{1}L^{2}T^{-3} |
Radiant intensity | M^{1}L^{2}T^{-3} |
Radiant power | M^{1}L^{2}T^{-3} |
Radiation pressure | M^{1}L^{-1}T^{-2} |
Rate of flow | M^{0}L^{3}T^{-1} |
Refractive index | M^{0}L^{0}T^{0} |
Relative Luminosity | M^{0}L^{0}T^{0} |
Resistance | M^{1}L^{2}T^{-3}A^{-2} |
Reasonant frequency | M^{0}L^{0}T^{-1} |
Reynolds number | M^{0}L^{0}T^{0} |
Rydberg constant | M^{0}L^{-1}T^{0} |
Specific gravity | M^{0}L^{0}T^{0} |
Specific heat | M^{0}L^{2}T^{-2}K^{-1} |
Specific resistance or resistivity | M^{1}L^{3}T^{-3}A^{-2} |
Specific volume | M^{-1}L^{3}T^{0} |
Speed | M^{0}L^{1}T^{-1} |
Stefn’s constant | M^{1}L^{0}T^{-3}K^{-4} |
Surface density of charge | M^{0}L^{-2}T^{1}A^{1} |
Surface potential | M^{0}L^{2}T^{-2} |
Temperature | M^{0}L^{0}T^{0}A^{1} |
Thermal expansion coefficient | M^{0}L^{0}K^{-1} |
Torque | M^{1}L^{2}T^{-2} |
Trigonometric ratio | M^{0}L^{0}T^{0} |
Velocity | M^{0}L^{1}T^{-1} |
Velocity gradient | M^{0}L^{0}T^{-1} |
Velocity of light in vaccum | M^{0}L^{1}T^{-1} |
Viscosity | M^{1}L^{-1}T^{-1} |
Volume | M^{0}L^{3}T^{0} |
Wave number | M^{0}L^{-1}T^{0} |
Wien’s constant | M^{0}L^{1}T^{0}K^{1} |
Work | M^{1}L^{2}T^{-2} |
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.
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.
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 × σ^{2}
Where,
Y is young’s modulus of elasticity.
σ is the strain.
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.
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 $$ Y = \frac{F L}{A \Delta L}$$
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 is given by
$$ G = \frac{\sigma_s}{\theta}$$
Where $$ \sigma_s$$ is tangetial stress and $$ \theta$$ is shearing strain.
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.
In SI, there are seven base ( fundamental ) units and two supplementary units.
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 |
Physical quantity | Unit | Symbol |
---|---|---|
Plane angle | Radian | rad |
Solid angle | Steradian | sr |
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.
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. [ M^{1}L^{2}T^{-2} ].
SI unit of energy is joule and CGS unit of energy is erg.
Newton’s gives three law of 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.
Time-rate of change of momentum is proportional to the applied external force. This is Newton’s second law motion
To every action there is always equal and opposite reaction. This is Newton’s third law motion.
A system of large number of particles in which the distance between any two particles remains fix throughout the motion is called rigid body.
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.
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.
If \(\mu\) and P are constant then idael gas equation becomes,
$$ V \propto T $$ This is the Charle’s law.
If \(\mu\) and V are constant then idael gas equation becomes,
$$ P \propto T $$
This is the Gay lussac’s law.
Basically waves can be three types:
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.
Waves on the water surface, light waves, wave generated on the string etc are the examples of transverse waves.
Sound wave propagated in air is the example of longitudinal 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.
These waves are associated with moving particles of matter like electrons, protons, neutrons, atoms, molecules etc.
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 [ M^{0}L^{0}T^{1}A^{1} ]
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.
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^{०}
∴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^{०}
∴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^{०}
∴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 = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}
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.
Then $$ \theta = 0^o or 180^o$$
So $$ \vec{A} \times \vec{B} = 0$$
For unit vectors
$$ \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$$
Then $$ \theta = 90^0$$
So $$ \vec{A} \times \vec{B} = AB \hat{n}$$
For unit vectors
\( \hat{i} \times \hat{j} = \hat{k}\) , \( \hat{j} \times \hat{k} = \hat{i}\) , \( \hat{k} \times \hat{i} = \hat{j}\)
$$ \vec{A} \times \vec{B} = ( A_y B_z – A_z B_y ) \hat{i} +
( A_z B_x – A_x B_z ) \hat{j} + ( A_x B_y – A_y B_x ) \hat{k}$$
Errors of measurement = True value of a quantity – Measured value of a quantity
Suppose, the measured value of quantity be A_{m} and the error in measurement be ΔA. Then the true value of the quantity can be written as
A_{t} = A_{m} ± ΔA
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 \( \Delta a_m\) 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 \( R_e\) is given by,
$$ \ R_e = \frac{\Delta a_m}{a_m}$$
Percentage error \( p_e\) is given by,
$$ P_e = \frac{\Delta a_m}{a_m} \times 100 \% $$
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 [ M^{1}L^{2}T^{-2} ]. SI unit of work is joule. CGS unit of work is erg.
Relation between joule and erg is
1 joule = 10^{7} erg.
In terms of rectangular components, work is
W = x F_{x} + y F_{y} + z F_{z}
If angle between \(\vec{F}\) and \(\vec{S}\) lies between 0^{0} and 90^{0}, then work done is positive.
If angle between \(\vec{F}\) and \(\vec{S}\) lies between 90^{0} and 180^{0}, then work done is negetive.
The work done is zero, if
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.
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 |