Torque

 Torque

Photo by Glenn Hansen on Unsplash

Introduction : We have seen small children playing top (bhawra) which rotates for some time when given a force. Similarly any rotational motion is associated with some force called as torque. Any object performing rotational motion require a rotational force. This torque is a rotational analogous of force. The force has the ability to produce turning effect or rotational motion for point mass. We will learn more in this topic.

Torque : The ability of a force to produce rotational motion is measured by its turning effect is called moment of force (torque).

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Mathematical Expression

Consider a rigid body free to rotate about an axis passing through a point 'O' and perpendicular to plane of rotation shown in fig. Force 'f' is applied at point P and 'r' is the perpendicular distance between axis of rotation and line of action of force.

τ = f x r

τ = f x r sin θ

where,

τ = torque

f = force applied

r = position vector

Try this experiment on your own.

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│<<<Impulse│    │Centre of mass>>>│

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Impulse

 Impulse

Introduction : When an object is in motion it is having momentum which is given by product of mass and velocity. The momentum specifies the amount of force actually require to bring a body at rest. If the body is moving with large velocity it has large momentum so large force is  required to bring the body at rest and thus change in momentum is obtained. This change in momentum is related to a quantity called as Impulse. In this section we  are going to learn more about Impulse.

Impulse : Impulse is mathematically equal to change in momentum of a body.

We know that when there is an external unbalanced force acting on a system there is a change in the momentum. This change in momentum is equal to Impulse. If unbalanced external force is acting for a finite measurable time, then we can write .

J = F.t ------(A)

SI unit of impulse is N s.

Dimension = [ L¹   M¹    T^-1 ]

If a body moving is with momentum P1 and after measurable time t it come to halt with momentum P2. Then change in momentum gives Impulse.

Mathematically,

J = P2 - P1

where .

J= Impulse 

P1= Initial momentum

P2= Final momentum

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Electrostatics XI

 Electrostatics XI

Charges at Rest

Charges : Every object in the universe is made up of small unit called as atom. Every atom consist of two type of charges viz. (Negatively charged electrons and positively charged Proton) as shown in fig above. This charges has different characteristics depending upon the state of rest and state of motion. When the charges are in motion they constitute electric current and when the charges is at rest it constitute static system. We are going to discuss about characteristics of charges when it is at rest in this chapter.

Try this experiment below

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Electrostatics : The study of charges at rest is called as Electrostatics. We have observed various phenomenon in our daily life (when the comb is rubbed with hair and brought near small pieces of paper, it attracts the paper. This is due to the static charge developed on the comb due to rubbing of comb.

Method of charging an object: There are two method of charging a conductor viz (a) Conduction method  (b) Induction method. Let us understand it in detail.

(A) Conduction method of charging : The method of charging in which charges are developed  on a neutral object due to their interaction (contact)is conduction method of charging. Let us understand it with an example.

Consider two Neutral object A and B .Object A is having 5 electrons and  5 protons as shown in fig above, similarly object B is having 4 charges each. When these object are brought together in contact and rubbed with each other electron are transferred from on object to another a shown in fig below. Let us consider 1 electron from A gets transferred to B, thus the equivalent charges on the body  A is due to only one Proton as effect of 4 electron is cancelled out by 4 protons whereas, The equivalent charge on body B is due to 1 electron as effect of electron is cancelled out by 4 proton. The diagram below is well illustrated to understand the concept.

Thus Body A developed 1 positive charge as it donates electron and body B develops 1 negative charge ad it accept one electron. As this charges are at rest ,the study of such charges are called as electrostatics.

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(B) Induction method of charging : The method of charging in which charges are developed  on a neutral object without their actual interaction (contact) is called as induction method of charging. Let us understand it with an example.

From the figure above we can see that initially, object A is neutral. A negatively  charged body is brought near the body  and as  soon as the charged object is brought near the body A , the negative charges begin to move away due to force of repulsion and positive charge is moved toward it due to force of attraction. If we connect a conductor to the body A and other end of the conductor is connected to earth all the negative charges flows in the earth  and the charges remaining on the body is only positive charges and body is said to be positively charged. The diagram below is well illustrated to understand the concept.

Conductor connected to ground

Induction method

 │Quantization of charge >>>│

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Change Your life: Every people in this world want to be successful. Each individual is abided by the law of nature. The success of each individual is measured in terms of hard work and consistency. The higher the hard work and consistency Read more.

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The power of thinking: Changing habit  is a cause of time, surrounding and thinking. Every individual have a superpower within them which is the gift of nature. When they know about this power and  utilize it in effective way, it leads to miracle Read more.

Be  a winner : Every individual is a precious gift by god. Every individual is bestowed with infinite talent within him/her by nature. The human brain is a storehouse of  precious collections and this collections is the source to be a winner. Every individual when born start learning new thing, apply it and try to be a best version of himself/herself. Read more

The Key principle  to learn anything  :Reading Books. Almost all the books in the world have solution for all the problems in the world. So reading books make you active consistence and solve all your problem. Read more.

The 80/20 principle : In our competitive world, each individual have a choice to live a better life, to have more than other. In order to make up with this demand they work harder and harder. But this technique do not yield the expected results, A student learning for 8 hours may  not obtain good grades in exam. 

You can learn any thing at very low cost from standford university than you might Think. Read More

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Inertial and Non inertial frame of reference

 Inertial and Non-inertial frame of reference

Photo by Sawyer Bengtson on Unsplash

Introduction : Inertial and Non inertial frame of reference is well understood by understanding state of motion and rest. In our day to day life we come across many experiences of accelerated motion like circular motion, where frame of reference is well understood. Frame of reference is in accordance with accelerated motion. As any state of motion or state of rest is dependent (relative)motion, it is convenient to consider frame of reference. To know or describe a motion you need to describe or define some reference. Such a reference is called a frame of reference.

Inertial Frame of reference : The system in which newton's first law of motion holds good is called as inertial frame of reference. It suggest that if there is no net force, there will be no acceleration and the object moves with constant velocity.

Eg: Consider a simple pendulum hanged on the roof of a train which is at rest. Now the pendulum is also at rest with respect to train. When the train is moving with constant velocity then the pendulum is also moving with constant velocity with 0 acceleration and train is said to be inertial frame of reference as there is no force experienced by the pendulum. As soon as train accelerate or decelerate, the pendulum experiences a force and now the frame of reference i.e train no longer remains inertial frame.

Non-inertial Frame of reference : The system in which newton's first law of motion does not holds good is called as non-inertial frame of reference. It suggest that if there is  net force, there will be an acceleration and the object moves with variable velocity.

Eg: Consider the same example given below, a simple pendulum hanged on the roof of a train which is at rest. Now the pendulum is also at rest with respect to train. i.e the system is in inertial frame but  as soon as   train accelerate or decelerate, the pendulum experiences a force and now the frame of reference i.e train no longer remains inertial frame. Thus now the train is said to be non inertial frame of reference.

<<<Linear momentum>>>│<<<Next Topic>>>

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Kirchoff's Law of Heat radiation

 Kirchhoff's Law of Heat radiation

Introduction : Kirchhoff studied the behaviour of heat radiation and based on that result obtained from emissive power and absorptive power he made a understanding that radiation depends on wavelength specific radiative emission and absorption by a body in thermal equilibrium. Note: All body when heated emits the same kind of radiation which they absorbs. Thus Kirchhoff arrived at an conclusion known as kirchhoff's law of heat radiation.

Kirchhoff's law of heat : The coefficient of emission is always equal to coefficient of absorption at any given temperature.

Mathematically,

a = e   -------(1)

but we know that e = E / E_b   -------(2)

From (1) & (2)

a = E / E_b 

E / a = E_b -------(3)

From equation no (3) Kirchhoff's law can be defined as At any temperature, the ratio of  the emissive power(E) to the coefficient of absorption(a) is always equal to emissive power of a perfectly black body (E_b ) at  the same temperature.

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Theoretical proff of  Kirchhoff's law

Consider a thermally isolated system which can be considered as uniform temperature enclosure. Two body A which is ordinary and body B which is perfectly black body have emissive power E and E_b respectively. Let body A has coefficient of emission (e) and coefficient of absorption (a). Both the body have same area and  let Q be the amount of radiant heat incident on both the body in  a given time and thus heat radiation absorbed by the body A  is given by ,

Q_a  = a Q    ---------(4)           [Q_a  / Q = a]

Q_a  =  Radiant heat absorbed by the body

a = Coefficient of absorption

Q = Total heat incident

Thus we know that,

 Heat absorbed by the body = Heat emitted by the body

a Q  = E -----(5)

For perfectly black body the heat radiation absorbed is given by,

Q_a  = a Q  

 ( for perfectly black body a= 1)

Q_a  =  Q 

Thus we know that ,

Heat absorbed by the perfectly black body  =  Heat emitted by the body

Q  = E_b  -----(6)

from (5) and (6) dividing

a = E / E_b

E / a = E_b 

Hence proved

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Wien's Displacement Law

 Wien's Displacement Law

Introduction: From the black body spectrum, it is clear that as the temperature goes on increasing, the wavelength corresponding to maximum emissive power goes on decreasing. The graph of wavelength and emissive power shifts toward left. Thus the above conclusion result in formation of very important result which was explained using wien's displacement law.

Statement: 

The wavelength for which emissive  power of a  blackbody is maximum is inversely proportional to the absolute temperature of the black  body.

Mathematically, 

λ_max  ∝  1/T

λ_max  = b  x  1 / T 

where b is called as wien's constant. 

b= 2.898 x 10^-3 mK.

This law is  also referred to as displacement law as the temperature increases the , the maximum intensity of radiation emitted by body gets shifted towards shorter wavelength.

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Heat Radiation

Absorption,Reflection and Transmission of heat

Introduction : When heat radiation is incident on a body, some part of heat radiation is reflected, transmitted and absorbed by the body. This amount of heat absorbed,t ransmitted and reflected depends on the coefficient of absorption, Transmission and reflection of heat from a body. Coefficient of absorption, transmission and reflection of heat radiation from the surface of the body depends upon the nature of material, area of material. 

Mathematically,

If Q is the total amount of heat incident on a surface of the body, Q_a is the heat absorbed by the body ,Q_t is the heat transmitted by the body and Q_r is the heat reflected by the body then

Q = Q_a + Q_t + Q_r -----(A)

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Radiation Spectrum

Black body spectrum

Try this experiment given below and learn how spectrum of heat radiation behaves

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The study of black body radiation in terms of wavelength  was carried out by Lummer and pringsheim maintaining the black body at different temperature. The system was studied by both of them at different temperature. The result was manupulated on  a graph of intensity of radiation versus different wavelength corresponding to that radiation intensity.

From the experimental curves, it was concluded that 

(1) Intensity of radiation  emitted increases with increase in wavelength.

(2) For a particular wavelength (λ_max) , the intensity of radiation emitted  is maximum and then decreases  with further increase in wavelength.

(3) Area under  the curve ( Eλ) versus (λ)  represent total energy emitted per second  per unit area by a perfectly black body.

Different Spectrum at different temperature are shown above in diagram.

We conclude that,

1) Energy distribution curves continues to be non-uniform.

2) Peak of ( Eλ) versus (λ) curve shift towards left, It means as temperature increases value of (λ_max) decreases.

<<< Ferry's black body>>> l <<<Wien's Displacement Law>>>

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Specific heat of a gas

 Specific heat of a gas

Introduction: It is very clear from previous topic that equal amount of energy is distributed among each degree of freedom according to equipartition of energy. We can make use of equipartition principle to calculate specific heat of a gas with the help of mayer's equation. This would give us a mathematical approach for calculating molar specific heat of a gas at constant volume (C_v)and constant pressure (Cp).

Monoatomic gas: Consider 1 mole of gas at enclosed in a container at constant temperature. monoatomic molecules have 3 degree of freedom. Thus average kinetic energy per atom 3/2 Kb T. Thus total internal energy per mole is given by,

U = 3/2 NA Kb T

U = 3/2 R T ----( NA Kb = R)

we know that 

C_v = U/ dT

C_v = 3/2 R T / dT

C_v = 3/2 R -----(1)

From Mayer's equation

R = ( Cp - Cv ) 

Cp = R +  Cv )  

Cp = R +  3/2 R)

Cp = 5/2 R -----(2)

from (1) and (2)

γ = Cp / C_v

γ = 5/3

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Ferry,s Black Body

 Perfectly black body

Introduction : As we already studied that total heat energy incident on a body is sum of radiation absorbed by the body, radiation reflected by the body and radiation transmitted by the body. Thus accordingly we arrive a equation Q= Q_a + Q_r =Q_t . Thus based on that a material can be good absorber, reflector and transmitter and material can be poor absorber , reflector , transmitter.

Let us consider an example below using an illustrative experiment 

Consider two body A and B exactly of same area and same temperature initially. Body A is polished with shiny silver and body B is polished with dark black as seen in the picture above. Both the system is  exposed to thermal radiation. Accordingly some part of heat is absorbed, reflected and transmitted by both the body depending on their respective coefficient of  absorption, reflection, transmission.

The above experiment conclude that the body which is polished with shiny material absorb very less heat and the body which is black painted absorb more heat. To maintain their temperature the rate of absorbed heat is always equal to rate of emission. Thus the body A which is poor absorber is also a poor emitter but good reflector. On the other hand the body which is good absorber is good emitter and poor reflector. 

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Ferry's Black Body

It is a double walled structured. Outer surface is polished with silver and inner surface is polished with lamp black. It has a small aperture for incident of radiation inside the body. There is a small conical projection in front of aperture to avoid reflection back of radiation via the same path. Thus the radiation entering the body cannot escape out and thus about 98% of energy is absorbed.

<<< Specific heat capacity>>> l <<<Radiation spectrum>>>

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Specific Heat Capacity

 Specific Heat Capacity

Photo by Quino Al on Unsplash

Introduction : We come across many phenomenon of heat in our day to day life.I t is our day to day observation that when we keep two different vessel, let say  A and B of copper and  aluminium of same mass respectively in heat for same amount of time having initial temperature same absorbs different amount of heat. we conclude that different material require different heat to make same change in temperature. The secret behind this is studied based on the observation of specific heat capacity.

Specific heat capacity : The amount of heat required to raise the temperature of 1kg mass of system by1℃ rise in temperature is called as specific heat capacity

Mathematically,

s = ΔQ / m ΔT

where s = specific heat capacity

ΔQ = Amount of heat absorbed or given

m = mass of the body

 ΔT = change in temperature

if m =1kg and  ΔT=1℃  then s = ΔQ 

Molar Specific heat capacity : The amount of heat required to raise the temperature of 1mole  of system by1℃ rise in temperature is called as specific heat capacity

Mathematically,

c = ΔQ / μ ΔT

where c = molar specific heat capacity

ΔQ = Amount of heat absorbed or given

μ = 1 mole

 ΔT = change in temperature

if μ =1mole and  ΔT=1℃  then c = ΔQ 

SI unit of specific heat capacity is J / mol ℃

cgs unit of specific heat capacity is erg /mol ℃

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Conservation of linear momentum

 Conservation of Linear Momentum

Lets have fun doing this experiment by our own.

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Introduction : Newton's first law and second law holds good with respect to law of conservation of linear momentum. As we know that linear momentum P is the product of mass and velocity, thus law of conservation is very useful in calculating  initial and final velocity of a system given one of the velocity as well as mass is known.

Statement: Law of conservation of linear momentum states that, when the resultant external force acting on a system is zero, the total linear momentum of the system of the body is conserved or constant.

Mathematical Expression :

Consider two sphere A and B making a head on collision. Sphere A and B have mass m₁ and m₂ with initial velocity u₁ and u₂ respectively. 

Before Collision

let u₁ > u₂

_{The total linear momentum (P) of the sphere before head on collision is given by}

_P1  = m₁ u_1 + m₂ u₂ -------(1)

During Collision

As u₁ > u₂ two sphere collide with each other and let 't' be the time of collision.By Newton's third law, The force is given by,

F_AB = - F_BA -------(2)

After Collision

 Let two sphere A and B separate and start moving in forward direction and their velocities becomes  V₁ and V₂ .After collision sphere A decelerate  and sphere B accelerate. i.e (V₁ < V₂)

_{The total linear momentum (P) of the sphere after head on collision is given by}

_P2  = m₁ v_1 + m₂ v₂ -------(3)

From Newtons Second Law,

 m₁ v_1 - m₁ u₁ = m₂ u_2 - m₂ v₂

m₁ u_1 + m₂ u₂ = m₁ v_{1 +} m₂ v₂

_{P1 = P2}

<<<Laws of Motion>>>│<<<Impulse>>>

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Degree of freedom

 Degree of Freedom

Introduction: We know that every molecule can move in space in three dimension i.e x , y , z axis. This coordinates are enough to specify it's location in a space. If a molecule is restricted to move in a plane then two coordinates are enough to specify it's location. Let us  consider a body which can move in only two direction in a plane i,e  x and y direction of cartesian coordinate system, and if it is in motion then its velocity component in x and y direction  ( v_x and v_y)is enough to specify its position and motion. Thus the number of direction in which molecule can move and number of motion it can perform relates the degree of freedom. In the above fig, a molecule can move in translatory direction and can perform rotational motion. This parameter is enough to describe its position and motion

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Gravitation

Kepler's Planetary Law of motion

Introduction: A german astronomer, Johannes kepler studied about planetary motion and formulated very interesting and informative behaviour of motion of a planet.

Let us see some interesting experiment and understand about planetary behaviour.

Perform it by your own .

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Kepler's First law : Every planet revolves around the sun in an elliptical orbit with sun situated at one of the focii. This is law of orbit.

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Kepler's Second law : The radius drawn from sun to any planet sweeps out equal area in equal interval of time. This is law of area.

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Kepler's Third law : The square of the period of revolution of the planet around the sun is directly proportional to the cube of semi-major axis of the elliptical orbit. This is law of period.

Mathematically,

If r is length of semimajor axis then, this law states that

 T²∝ r3

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Fact About Ellipse

Ellipse is the locus of the points in a plane such that the sum of their distances from two fixed points, called the foci, is constant. You can draw an ellipse by the following procedure. 

1) Insert two tacks or drawing pins, A and B, as shown in the figure into a sheet of drawing paper at a distance ‘d’ apart. 

2) Tie the two ends of a piece of thread whose length is greater than ‘2d’ and place the loop around AB as shown in the figure.

 3) Place a pencil inside the loop of thread, pull the thread taut and move the pencil sidewise, keeping the thread taut. The pencil will trace an ellipse. 

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Be  a winner : Every individual is a precious gift by god. Every individual is bestowed with infinite talent within him/her by nature. The human brain is a storehouse of  precious collections and this collections is the source to be a winner. Every individual when born start learning new thing, apply it and try to be a best version of himself/herself.Read More

Change Your life: Every people in this world want to be successful. Each individual is abided by the law of nature. The success of each individual is measured in terms of hard work and consistency. The higher the hard work and consistency , the higher is the rate of success. For being a successful person you(each individual)should constantly work on being a unique you(i.e Being different from others). Read More

The 80/20 principle : In our competitive world, each individual have a choice to live a better life, to have more than other. In order to make up with this demand they work harder and harder. But this technique do not yield the expected results, A student learning for 8 hours may  not obtain good grades in exam.Read More

Learn Game Development :  Computer as well as mobile Game had been on high demand nowadays and many gaming industries around the world face shortage of developers. Since many years Gaming industries have been taking over the market by proving game for android as well as windows phone. It has been a good source of earning as well as learning the behaviour .Read More

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