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True Discount and Banker's Discount problems tricks in Hindi | fast track formulae for problem solving.


TRUE DISCOUNT 1). वास्तविक बट्टा = मिश्रधन - वर्तमान धन
2). यदि ब्याज की दर r% वार्षिक , समय t व वर्तमान धन (pw) है, तो वास्तविक बट्टा = PW×r×t/100
3). यदि r% तथा समय t के बाद देय धन A है, तो तत्काल धन pw = 100×A/(100+r.t)
4). यदि देय धन A पर r% व समय t दिए गए है, तो वास्तविक बट्टा = A.r.t /(100+r.t)
5). यदि किसी निश्चित समय के पश्चात निश्चित वार्षिक दर पर, देय धन पर वास्तविक बट्टा (TD) व समान समय व दर के लिए साधारण ब्याज (SI) हैं, तो देय धन A =      SI × TD/(SI - TD)
6). यदि t समय पश्चात r% वार्षिक दर पर, देय धन पर वास्तविक बट्टा (TD) व साधारण ब्याज (SI) है , तो -  SI - TD = TD×r×t/100
7). t वर्ष बाद r% चक्रवृद्धि दर पर देय धन A का तत्काल धन (PW) = A/(1+r/100)^t   व वास्तविक बट्टा = A - PW
BANKER'S DISCOUNT1). महाजनी बट्टा = शेष समय (समाप्त न हुए समय) के लिए बिल पर ब्याज = बिल की राशि × दर × शेष समय /100
2). महाजनी लाभ = महाजनी बट्टा - वास्तविक बट्टा
3). यदि बिल का मान / अंकित मूल्य A है, समय t व दर r% है, तो महाजनी बट्टा = A×r×t/100

POISEULLIE'S LAW, Understanding the basic concepts and it's mathematical derivation//

So today in "laws of nature" we are going to talk about a very important and interesting concept of hydrodynamics , that is poiseullie's laws.
So going further it is necessary to understand the underlying concept of poiseullie's laws in very simplest  form.
In poiseullie's laws fluid motion should be streamline, through any pipe.
Explaination of  Poiseullie's laws and it's mathematical derivation//
So let's starts:
We are starting the concept with a question that in what manner a viscous fluid flows through a cylindrical pipe?
In any cylindrical pipe when a viscous fluid flows then they follow a convex meniscus,but why? It is because, we know that a streamlined fluid has several stream of fluid.

Then some streams been near the wall of the pipe and some stream is far from the wall, In the cylindrical pipe the longest distance from the wall is the centre of the pipe.
And we know that the pipe is cylindrical then the shape of water which are flowing through it, must be in the shape of cylinder.
Then we can conclude that every fluid lamina in the pipe are in the shape of cylinder.there are infinite laminae between the centre of the pipe and it's wall.

And as fluid starts flowing through the pipe then friction also starts between the wall of the pipe and the lamina which is nearer to it. And this friction opposes the velocity of the fluid, and at the wall velocity of fluid is taken as zero, because at this point maximum friction is applied.
And friction occurred between the curved surface area the pipe and fluid lamina.
Due to this friction a opposing forces generated and this force is termed as laminar force or viscous force.

And in the pipe each lamina opposes the velocity of each lamina, first lamina opposes second lamina, and second opposes the third and so on. And this opposition started from the wall, so the stream which is nearer the wall has minimum velocity, and as distance increases from the wall velocity of the fluid also increases, and become maximum at the centre, it means it's velocity is effected by the change of radius ml and same occurred from the other sides of the wall, so this form the convex meniscus of the fluid in the pipe.

And this is not only for cylindrical body, this laminar flow held in all types body in which fluid is flowing. This gives a convex form as given below.
Explaination of  Poiseullie's laws and it's mathematical derivation//
So now it's time to talk about the poiseullie's laws:
First of all we have to know that poiseullie's laws measures what?
Poiseullie's laws measures only the rate of flow of volume of fluid per second in the pipe.
Going further let's give a sight on, at what factors can volume rate depends:

If we think little then we observe that flow rate depends directly on the pressure difference across the pipe, because if pressure difference is small the flow of fluid become slow, but if pressure difference is large then flow rate increases, we have all seen it in daily life.

Second is, its depends directly to the radius of the pipe, how? Think yourself that what would happen if radius increases, if radius increases then it's cross sectional area also increases, if it's cross sectional area increases then fluid get more space to travel and suffer a little friction, if friction is small then flow rate increases automatically.

It is also a daily basis phenomenon , if any road is narrow then it suffer a high traffic, but a broader road is free from traffic. Same is here.
And third is flow rate is inversely proportional to the viscosity, larger viscosity larger friction then larger opposition and then smaller velocity then smaller flow rate.

And fourth is it is inversely proportional to the length of the pipe, how? If length increases then time of friction to the wall also increases, and friction increases then same as above.
So now we are going to derive a formula for rate of flow of fluid volume per second.


Let's take a cylinder of radius R and length L as shown below,
And flowing a fluid of viscosity η , and dr is the small distance between two cylindrical lamina. For understanding see diagram below.
Explaination of  Poiseullie's laws and it's mathematical derivation//
When a fluid is in the pipe then the force of viscosity generated due to the friction between the two lamina is given as below:
           F = -ηA(dv/dr)
Where A is the curved surface of the lamina with radius r and length L that is;
              A = -2πrL
And dv/dr is the velocity gradient, means velocity changes with change in radius.
Here sign is negative because this force opposes the velocity of the fluid.
Then putting all the values F become
           F= -2πηrL(dv/dr)
And we know that fluid is flowing through the pipe due to pressure difference;
                        P = P1-P2
This pressure can be expressed in external force per unit area, then force is given as;
                      F = Pπr^2
In equilibrium condition, when external force became equal to the laminar frictional force that is viscous force.
Equating both forces we get
                      ✓ Here we take only magnitude
                  Pπr^2 = 2πηrL(dv/dr)
After simplification we get,
                  dv/dr = Pr/2ηL
Then.     dv = (Pr/2ηL)dr    ,  now integrating
Both sides from r to R, we get
        Int(dv) = (P/2ηL).int(r to R) rdr
On solving we get
            v = [P/4ηL](R^2-r^2)

Now applying equation of continuity, we get
             V = A1v1=A2v2.     Where V is volume

Then for a small cross sectional area dA we get small volume of fluid that is dV
       dV = dA×v.      Putting all values, we get
     dV = 2πrdr× P(R^2-r^2)/4ηL
On solving we get
                  dV = Pπ(r.R^2-r^3)dr/2ηL
Again integrating both sides from 0 to R because we want to find whole volume of fluid in the pipe. Then we get
   Int(dV) = (Pπ/2ηL).int(0 to R)[r.R^2-r^3]dr

On solving we get
         V = (Pπ/2ηL)[(R^2.R^2)/2 -(R^4/4)]
         V= (Pπ/2ηL)[(R^4/2) - (R^4/4)]
On solving we get
                     V = PπR^4/8ηL
This is the fluid which are flowing per second through the pipe.



Boat and stream short-tricks in Hindi | Fast track arithmetic formulae for competitive examination.

BOAT AND STREAM (नाव एवं धारा)
1). यदि शांत जल में नाव या तैराक की चाल x किमी/घंटा व धारा की चाल y किमी/घंटा है, तो धारा के अनुकूल नाव अथवा तैराक की चाल = (x+y) किमी/घंटा
2). धारा के प्रतिकूल नाव अथवा तैराक की चाल = (x-y) किमी /घंटा
3). नाव की चाल = (अनुप्रवाह चाल + उद्धर्वप्रवाह चाल)/2
4). धारा की चाल =  (अनुप्रवाह चाल - उद्धर्वप्रवाह चाल)/2
5). यदि धारा की चाल a किमी/घंटा है, तथा किसी नाव अथवा तैराक को उद्धर्वप्रवाह जाने में अनुप्रवाह जाने के समय का n गुना समय लगता है,(समान दूरी के लिए), तो शांत जल में नाव की चाल = a(n+1)/(n-1) किमी/घंटा
6). शांत जल में किसी नाव की चाल x किमी/घंटा व धारा की चाल y किमी/घंटा है, यदि नाव द्वारा एक स्थान से दूसरे स्थान तक आने व जाने में T समय लगता है, तो दोनो स्थानों के बीच की दूरी = T(x^2 - y^2)/2x km
7). कोई नाव अनुप्रवाह में कोई दूरी a घंटे में तय करती है, तथा वापस आने में b घंटे लेती है, यदि नाव कि चाल c किमी/घंटा है, तो शांत जल में नाव की चाल = c(a+b)/(b-a) km/h
8). यदि शांत जल में नाव की चाल a किमी/घंटा है, तथा वह b किमी/घंटा की चाल से बहती हुई नदी में गत…

Emergence of British East India Company as an Imperialist Political Power in India

Dynamically changing India during early eighteenth century had a substantially growing economy under the authority of Mughal emperor Aurangzeb. But after his demise in 1707, several Mughal governors established their control over many regional kingdoms by exerting their authority. By the second half of eighteen century, British East India Company emerged as a political power in India after deposing regional powers and dominating over Mughal rulers. The present article attempts to analyze the reasons for emergence of British East India Company as an imperial political power in India and their diplomatic policies of territorial expansion. In addition to this, I briefly highlighted the Charter Acts (1773, 1793, 1813, 1833 and 1853) to trace its impact on the working process of Company.Establishment of East India Company in India

In 1600, British East India Company received royal charter or exclusive license…

Three dimensional geometry (part-1) | study material for IIT JEE | concept booster , chapter highlights.


In the following diagram X'OX , Y'OY and Z'OZ are three mutually perpendicular lines , which intersect at point O. Then the point O is called origin.
In the above diagram X'OX is called the X axes, Y'OY is called the Y axes and Z'OZ is called the Z axes.
1). XOY is called the XY plane. 2). YOZ is called the YZ plane. 3). ZOX is called the ZX plane.
If all these three are taken together then it is called the coordinate planes. These coordinates planes divides the space into 8 parts and these parts are called octants.
COORDINATES  Let's take a any point P in the space. Draw PL , PM and PN perpendicularly to the XY, YZ and ZX planes, then
1). LP is called the X - coordinate of point P. 2). MP is called the Y - coordinate of point P. 3). NP is called the Z - coordinate of the point P.
When these three coordinates are taken together, then it is called coordinates of the point P.

A detailed unit conversion table in Hindi.

CENTIMETRE GRAM SECOND SYSTEM (CGS)1). MEASUREMENT OF LENGTH (लंबाई के माप) 10 millimeter = 1 centimetres10 centimetre = 1 decimetres  10 decimetre = 1 metres 10 metre = 1 decametres 10 decametres = 1 hectometres 10 hectometres = 1 kilometres 10 kilometres = 1 miriametresMEASUREMENTS OF AREAS ( क्षेत्रफल की माप )  100 millimetre sq. = 1 centimetre sq.
 100 centimetre sq. = 1 decimetres sq. 100 decimetres sq. = 1 metre sq. 100 metre sq. = 1 decametres sq  100 decametres sq. = 1 hectometres sq. 100 hectometres sq. = 1 kilometres sq. 100 kilometres sq. = 1 miriametres sq.
MEASUREMENTS OF VOLUME ( आयतन की माप) 1000 millimetre cube. = 1 centimetre cube.
 1000 centimetre cube. = 1 decimetres cube. 1000 decimetres cube. = 1 metre cube. 1000 metre cube. = 1 decametres cube. 1000 decametres cube. = 1 hectometres cube. 1000 hectometres cube. = 1 kilometres cube. 1000 kilometres cube. = 1 miriametres cube.
MEASUREMENTS OF VOLUME OF LIQUIDS  (द्रव्य के आयतन का माप) 10 millilitre=…

THE GENERAL THEORY OF RELATIVITY | A Unique way to explain gravitational phenomenon.

Today we are going to talk about a very important and revolutionary concept that is THE GENERAL THEORY OF RELATIVITY.
This theory came into existence after 10 years of special theory of relativity (1905), and published by Albert Einstein in 1915.
This theory generalise the special theory of relativity and refines the Newton's laws of universal gravitation.
After coming this theory people's perspective about space and time has been changed completely. And this theory give a new vision to understand the spacetime geometry.
This theory gives a unified description of gravity as a geometrical properties of space and time.
This theory helps us to explain some cosmological phenomenon that is ,

* why small planets revolve around the big stars?
* Why everything in this universe is keep moving?
* Why mostly planets and stars are spherical in shape?
* Why does gravity create?
* Why does time become slow near the higher gravitating mass. Ie. Gravitational time dilation.
And gravitational…