 okay let's take a few numericals now write down the register you have the measurement of voltage and current voltage is measured as 100 plus minus 5 volts okay and current is measured as 10 plus minus 0.2 ampere okay you need to find the error in the resistance. Find the error in the calculator resistance. Have you done derivatives? Not yet. No knowing is different before but it is not done in center right now. Okay so when you are done with derivative then you will come to understand there using derivative all of these things can be done easily. You will take log both sides and differentiate when you take log the variable goes in denominator and delta comes in the numerator okay but then right now till you have not done derivative we will not do it right now. We will do it later on. Solve this question. 7-0? Yes sir. It cannot be 7-0. What is the true value of the resistance? 10 by 10 or 100 by 10. So any error more than 10 is something wrong. 0.07. Yeah 0.07? No 0.07 is correct. Yes absolute is only. I have asked. What is the error? You have to calculate absolute error. If I am asking it relative error I will specify. What is the relative error? Resistance ratio between voltage and current. All of you remember right? So delta r by r equal to delta v by v plus delta i by i. What is delta v by v? Delta v is how much? Error in v 5? 5 by 100. Sorry. 2 by 100. So this is 7 by 100. So percentage error is 7 percent of the resistance. Yeah. Okay and the value of r is 100 by 10. 0.7. 0.7 ohm is the absolute. Okay do the resistance if resistance is connected in parallel. Yeah. What is that? 1. Now if I ask you to find out the error in r equivalent given error in r1 and r2. Will you be able to? Yes sir. But done this kind of thing right? We have only done multiplication addition and exponents. I think you can do that. Yeah. But this is summation of inverse of that. Yeah. It is still R1 by R1 plus R2. Okay. So r equivalent is you are saying R1 plus R2. Then what you will do? This is multiplication and division together. Yeah. Multiplication, division and addition together. So we have not done that. We only have done only addition, only subtraction. And you can have multiplication, addition, subtraction together. And the same expression summation as well as multiplication. That we have not done. But it doesn't mean that I don't have error. There is an expression for this also. The way you find expression being help of derivative. Okay. In your book they have directly given the expression. They have told that in this situation the error formula is this. Okay. Let me just quickly tell you how they have gotten. Those who know the derivative can appreciate. But then you don't need to know. No. I am just telling you how it is coming. If you differentiate it with respect to any variable, let's say variable is X. If you differentiate 1 by R equivalent, you will have 1 by R equivalent minus 1 by R equivalent square into dr equivalent by dx. It is equal to minus 1 by R1 square dr1 by dx minus 1 by R2 square dr2 by dx. Okay. This dx dx is gone. So minus is also gone. So you have dr equivalent by R equivalent square is equal to dr1 by R1 square plus dr2 by R2 square. So if your error is very small, let's say error is delta R. You can write then you can say that delta R equivalent by R equivalent square is equal to delta R1 by R1 square plus delta R2 by R2 square. Okay. Actually in physics there will lot of times you know that we have to come back to the same chapter after sometime if you are preparing for J. Because these kind of thing you will not understand if you don't know what is derivative. So when you learn derivative, we will come back to this chapter little bit, talk about this kind of scenario. And then this chapter is done. Similarly when we do kinematics, you need to know what is integral calculus and all those things that will come up. Okay. So once you are done with integral calculus basics of it, we have to again come back on kinematics and do it. So right now how many of you have understood this? I got it with me. So generally we don't cancel dx dx. Yeah, I got it. Why not? Because only integration we treat d something as a block. What is drx by dx? Dividing drx divided by dx. Yes or no? Why can't you cancel dx? But it's like canceling a multiplication science. We have 2 to 3 is 3 to 4. It's like canceling multiplication multiplication. Oh my god. What does it say 2 to 3 is equal to 3 to 4 you are saying? It's like canceling out the 2 multiplication science. It's an operator. So what exactly are you doing? First of all these are not 2 equal. No, I mean if you have something there. You can do 3 into 2. Are you saying operator? Yeah. It is operator. No, it is not operator. It is simply what is happening here is this. You are changing x little bit. Yeah. How much dx? dx is change in x. Yeah. If you change x by dx amount, how much change r has which is dr? Derivative simply a ratio of change in r with change in x. That's all. Okay. Derivative is a method. That's all. Okay. What do you think dx is square by dx is 2x? It is derived. Okay. When this delta x tends to 0, it becomes 2x automatically. I hope I am clear. If you change x little bit, what will x square become? x plus delta x, delta x plus minus delta x the whole square. This is what it will become. So if delta x is tending to 0, this will tend towards you can take x square outside. It will become 1 plus minus delta x by x the whole square then binomial approximation. It will become 1 plus minus 2 delta x by x only when delta x is very small. Delta x is close to dx okay. So change in x square is how much? Change in x square is how much? It has become x square plus minus 2x dx. That's all. Right. So change in x square is this 2s dx. If you divide it from dh it becomes 2x. So it's not operator. It is a method. So you can call it loosely an operator. But this operator is not same as multiplication sign. Multiplication is a sign. But then of course if 2 into 3 is equal to, what it will be equal to? 1 into 6. Then you can't cancel 1 and 2. Okay. Anyways we can discuss all this later on. Right now you have less understanding of calculus. But definitely you can cancel out dx like that. Alright guys next question. Next question please focus here. Okay time period of oscillation of a pendulum. You know what it is? T is equal to 2 pi under root l by g. This is the formula. The measured value of length is 20.0. Measured value of length is 20.0 centimeter. Okay. It is known to 1 mm accuracy. What does it mean? 1 mm accuracy. It means that the random error is 1 mm plus 1 mm minus 1 mm plus 0.5 mm minus 0.5 mm but whatever it is it fluctuates plus minus 1 mm. So length is 20.0 centimeter known to 1 mm accuracy and time for 100 oscillation. Time for 100 oscillation is found to be 90 seconds. This time is calculated using a wristwatch of 1 second accuracy. Accuracy is 1 second. So definitely you can't calculate the 1 time period with accuracy of 1 second. I cannot calculate less than 1 second. So that is why I am calculating time for 100 oscillation. Understand? So this is the scenario. You need to find out the accuracy. You need to find accuracy in determination of g. When I say accuracy also means that I am asking you to find the error in finding errors. Same meaning. Although they are opposite. The hint is the formula is for 1 oscillation and formula is for time period. Should I do it? Pi is pi. Pi is pi but pi squared is 10 roughly. Remember this. Pi squared is 10. It helps. But the error in pi is 0. It is a constant. Can I do it now? Even tell me percentage error or any error. Root will not come. So getting root something wrong. Done? How much? 1 by 400. 1 by 400. Square and all the other time. Should I do it now? Hit it in a form of y is equal to function of whatever the measured quantity. For what you are calculating the error? What are the quantities you are measuring? Pi squared and t and l on the right hand side. Modify the expression. So you will get t squared equal to 4 pi squared l by g. So g will be what? Pi squared l by t squared. This is what you will get. Problem is that this is the equation for 1 time period. For 100 time period what will be the equation? 100 times. 100 times t should be equal to 200 pi l by g. You are not measuring t. You are measuring 100 times t. So you modify the formula by writing this as t100. Please ignore this. First you modify the formula in terms of what you are measuring. Understood? You are measuring 100 times of t. You are not measuring t. All of you understood? So now you get the value of t in terms of t100. T100. Time for one oscillation is 9 tenth of a second. No, don't do that. You are not measuring 1 tenth of the time period. You are measuring the time period of 100 oscillation. You should write in terms of what you are measuring. I can understand your feelings but that is not how it is done. You should write this error of 1 second is on the error oscillation. Then only you can use the error as 1 second under root l by g. So t100 square equal to let's say 200 pi square into l by g. So g will become equal to 200 pi the whole square l by t100 square. Now how to find the error in g? This is a division of two measured quantities. So relative error in g will be what? Delta g by g will be equal to pi square which is what? 0. So error in then this is equal to delta l by l to delta t100 divided by t100. At this stage I should not say it is wrong. Let me solve it. Okay so delta g by g will be what? Delta g by g. How much is delta l which is 0.1 centimeter? Both units should be same. Delta l and l both units should be same. So 0.1 divided by 20 that's also it is fine. Again it will be same itself. Plus two times what? Delta t is how much? One. One second of accuracy divided by so when you substitute all of 0.032 so the percentage error is 3 percent. You discard the third decimal point. It can't be more accurate than the reading itself. Okay. Any doubt here? This is from your NCIT solve exercise. Wait so we were supposed to find relative. Yeah I thought we had to find delta g. Delta g will be g into 0.032 right? Yeah that will be complicated. Oh g is 10. But how can you use okay the true value of g is close to 10 or it will be given if it is not given you can. I thought we should calculate from. But did you get 0.032? You have to anyway calculate. You got that. So yeah even I got 0.032 but then I thought for g we are going to have to substitute everything there. G is always there. No but shouldn't why should we take it? Yeah you can calculate g from here. You can calculate g. You can put this as 90. You can put L as 20 centimeter. You will get the value of g. So it is 10 to the power 4 divided by 81. So that was weird. Terrible. Right so they mean to say 0.1. Relative but if you have to find the absolute error first you have to calculate g from this expression by stop talking. First you have to substitute the value of 100 times t which is 90 second over here 0.2 meters over here to get the value of g. And then that you use here to get delta g.