 So, this is another problem video on chapter number 1 units and dimensions we will be starting with NCRT problems. NCRT is one of the most standardized textbooks and I have seen some students go like what is the point of doing NCRT we are preparing for advance and all that but you have to understand that everything every competitive exam in India including need J mains and J advanced have their foundation built over the NCRT okay. So, never underestimate NCRT and you have to understand that this is one of the most important books and also the depth of knowledge in NCRT is very high okay. Obviously, it's not meant for reading. Reading in that sense you know you cannot just go and start reading NCRT for that matter because the explanation in that is very summarized. So, it is not useful in understanding some stuff okay but once you understand with the help of your teachers any topic then it is mandatory that you do a question at least some questions from NCRT to you know understand the direction in which the content is flowing okay. So, starting our journey from NCRT problems that was do some NCRT problems okay. So, let's start with this question okay and as physics students it is very highly expected of you to make diagrams. So, every time you see me solve a question I'll always try to make diagrams for it. So, this is an example from NCRT the moon is observed from two symmetrically opposite points A and B on earth and the angle theta subtended by the moon by the two directions of observation is 1 degree 54 1 degree 54 minutes okay given given the diameter of earth to be something into 10 to power 7 compute the distance of the moon from the earth okay. So, the question says key there are two stations on earth which are diametrically opposite of each other okay and they start observing the moon from their location like this cool and the angle that these observatory subtend on the moon is about 1 degree 54 minutes okay and we have to essentially find out what is the distance of the moon from the earth what is the distance from the moon to the earth okay. So, I told you if the distances are very high the curvature would almost look like a straight line and we are entitled to use or we can possibly use the formula which says key you know the L by R where L is the arc length divided by the radius is equal to theta okay and this is the definition of radiant one radiant is nothing but the arc length divided by the radius okay. So, we are entitled to use this definition here but our problem the first problem you'll notice here is that the angle is in degrees not in radiant so my first job would be to convert this degrees into radiance okay so if I think about it 180 degree is 5 okay I just have to figure out how much is 1 degree 54 minutes in terms of radiance in terms of radiance okay so the thing here is that first you have to convert this thing into completely into degrees first because I cannot deal with 1 degree 54 minutes so what I'm gonna do is write this like this 54 minutes divided by 60 minutes is one it's going to be it will convert this 54 minutes into degrees okay so this is how much this would be 69 or 54 6 10 or 60 so that is 0.9 degrees so that would mean key this 1 degree 54 minutes is actually 1.9 degrees okay so 1 degree and 54 minutes is nothing but 1.9 degrees I have to find out how much this is in radiance okay so I'll use the unitary method okay this unknown value is nothing but 1.9 degrees divided by 180 degrees into pi okay and this will come out to be let's check out let's let's use the calculator here let's use the calculator here one second 1.9 divided by 180 okay and this multiplied by 3.14 that gives me 0.0331 radiance okay so I found out this angle in terms of radius now all I have to figure out what is the arc length here in this case this is the arc length and this is the radius this r value is what we have to find out and because these this distance is very very high whatever this value of r is that will be same as the distance from the moon to the earth okay so all I have to do is plug in the value of L by r in this equation I have to just plug in the values so from here r would be equal to L by theta L is nothing but it's given in the question the diameter of the earth is L 1.276 1.276 into 10 raised to power 7 meters is L and theta we just found out is 0.0331 radiance okay so this will when I do this calculation let me do this calculation okay 1.276 that divided by 0.0331 okay gives me 38.5 gives me 38.5 this gives me 38.5 into 10 to power 7 meters so if I make it into standard form that would be 3.85 into 10 to power 8 meters okay so this is the distance of the earth from the moon okay this is the distance of the earth from the moon so this is how you do it if you have any doubts put it in the comments let's move on to the next question okay so let's do this another question and in this question they're asking us to measure the time period of an oscillation of a simple pendulum okay and there are a bunch of readings that are taken for the oscillations of this pendulum and I'm going to note down all this measurements okay so the first measurement was taken to be 2.63 seconds second measurement is 2.56 seconds third measurement is 2.42 seconds fourth is 2.71 and the fifth is 2.80 okay so what I have to do is I have to find out what is the also let me just highlight them I need to find out what is the absolute errors what is the relative error and what is the percentage error okay so error is what the deviation from the true value but how do you find out the true value from given set of measurement the answer is by taking the average true value is nothing but what is the value that we are going to assign it to that particular measurement so I got five readings but if someone asked me what is the time period what is the value that I'm going to give to them is the true value okay so to find out true value all I have to do is this so I'm gonna call this AT T stands for true value this is equal to a1 plus a2 plus a3 plus a4 plus a5 by 5 okay so this is going to be the true value in this case let me all and let me add all of these values in a calculator okay so all the values are 2.63 plus 2.56 plus 2.42 plus 2.71 plus 2.81 this comes out to be 13.12 seconds okay and all I have to do is divide this by 5 okay so divide this okay so one second this divided by 5 gives me 2.62 okay 2.62 seconds so this is the true value that I was looking for 2.62 seconds now what I can do is I can find out the individual errors in each of these measurements by subtracting these values okay so a1 minus AT will give me exactly how much error is present present in a1 so what I'll do is 2.63 minus 2.61 2.62 that means the error in the first measurement is going to be equal to 0.01 okay similarly what is the error in the second measurement this is going to be equal to 2.62 minus 2.56 okay this would be 0.06 sorry I think I made a mistake here this should be 2.56 minus 2.62 okay remember it's the measured value minus the true value so that means the error in this is minus 0.06 very very important stuff here okay what is a3 minus AT what is the error in the third measurement it is 2.42 minus 2.62 that means an error of minus 0.2 okay minus 0.2 let's do the next one a4 minus AT is going to be equal to 2.71 minus 2.62 makes it 0.09 okay and the final one is a5 minus AT which is going to be equal to 2.81 minus 2.62 that means this one would be 0.19 okay 0.19 cool so what should we do next I found out the individual errors in all of them but to find out the absolute error I have to add all these errors together and when I say add I have to add their absolute values because we are looking for the absolute error here okay so I have to take modulus of all these errors add them together and then divide it by five that would be the absolute error in this okay so what I have to do is I'll have to add 0.01 add 0.06 add 0.2 add 0.09 and add 0.19 so there is one other statement that I always gave in class key errors if you make two mistakes they will they will always add up okay mistakes are not something that you can subtract them even if you like if two quantities are there in your subtracting them the error in them will still be added together because although they are subtracted mistakes will never get subtracted if the more mistake you make in measuring something all those mistakes will increase the overall mistake in the final answer so even though the quantities are getting subtracted the errors will never get subtracted they will always get added up okay so there is one thing that you have to remember so let me add up all these values here also I have to divide this by five because there are five errors here okay so let me just quickly put this in my calculator 0.01 plus 0.06 plus 0.2 plus 0.09 plus 0.19 this is equal to 0.55 0.55 all these errors added up together will be 0.55 when I divide this by five I'll get 0.11 as the absolute error so this is the value of absolute error if I want to find out relative error relative error okay that is just absolute error which is 0.11 divided by the true value so how much error is present in what value okay so the true value was 2.60 so this fraction is nothing is nothing but the relative error okay so 0.11 divided by 2.62 yields 0.042 0.042 okay so this is the relative error if I want to find out what is the percentage error again it is very very simple okay this is just 0.11 divided by 2.62 into 100 this answer will come in percentage which would be 4.2 percent error okay 4.2 percent error so I got all the values we have relative error we have percentage error and we have absolute error okay again if you have any doubts please let me know in the comments okay let's do some another question okay in this next question they are asking us what is the relative error in z is if z is given by a raised to power 4 b raised to power 1 by 3 divided by c d raised to power 3 by 2 so this and this comes under the topic of combinations of error so that means if one function is essentially expressed as a you know as a combination of other functions how would you find the net combined error in z if I know the errors in a b c and d okay so that is what the question is also asking if we know the relative error in a b c d what is the error in z that is the question here okay so let me remind you the formula that we use if we have z as a raised to power some m then delta z by z is m times delta a by a that is what I'm going to use in this particular problem this is the formula I'm going to use okay so here this is a combination of four times so what I'm trying to find out here is the relative error delta z by z this is equal to so for the first one here a 4 the relative error would be four times delta a by a again where did this come from this this example this formula okay now the relative error in b will get added with this value okay so plus one third of delta b by b okay then plus delta c by c because the power for one for c is just one and at the end it will be plus 3 by 2 delta d by d now some people will be asking he said this the c and d are divided won't the error get you know subtracted some people might have that doubt for them the answer is very very simple no matter what happens in the function multiplication division or anything mistakes do not subtract okay mistakes only add up okay so if you make a mistake in calculating a and the mistake in calculating d no matter what your formulas is if you're multiplying a and d or dividing a by d these mistakes will always get added up that's why we're putting a plus sign irrespective of the fact whether the value is in the numerator or in the denominator okay so this is the answer for this question let us move on to the next question okay so in this question there are two registers okay the first register has a resistance of 100 plus minus 3 ohms so what does this this notation means that the resistance is 100 ohms and an error of three absolute error of three ohms is present in in our okay the error in our one is three ohms and its value is hundred ohms similarly the resistance to here resistance to here is 200 ohms and the error in the second register is four ohms okay so this so I've seen some people say is a resistance one is hundred plus minus three ohms so the resistance is maximum value is three hundred and three ohms and the minimum value is 97 ohm I've I want to say that that particular analysis is not correct okay don't don't don't think he the resistance is maximum value is 103 ohm the we have the actual value of the resistant it is hundred ohms okay and treat the three ohms separately that three ohms is nothing but the but the error which is generated in calculating our one this this doesn't mean you have to add hundred plus three or hundred minus three that particular thing you don't have to do okay treat them separate as separate quantities now if they're what they're asking is I if I arranged these two registers in the in a series combination what would be the net resistance so we know key we know this formula key if let's say z is equal to a plus b okay then the error in z will also be error in a plus error in b okay so when I combine r1 and r2 in a series combination in a series combination our net will be what our net will be r1 plus r2 it will be r1 plus r2 okay so from here okay so the value itself the rnet will be how much it will be hundred plus 200 okay this will be 300 ohms okay I just found out what is the rnet just from r1 and r2 okay now I need to find out what is the error in this the rnet okay what is the error in the rnet again I will write this as delta r1 plus delta r2 this I got from where from this if z is equal to a plus b delta z will be equal to delta a plus delta b formula can be found in combination of errors okay so from here I found out so that means the error in rnet will come out to be three ohms plus four ohms which is seven ohms okay so how will I write the final answer the final answer I will write like this so the final if you keep the resistances in series the net resistance is 300 ohms plus minus seven ohm again I'm going to repeat one statement that I said before this doesn't mean our resistance is 307 or 293 you don't have to do 300 plus 7 or 300 minus 7 300 is the value of the resistance measured resistance and seven ohm is the is the error which is present in this calculation okay so treat them separately I'm saying this one more time the next part of the question asks see what what will be the net resistance if we arrange it in a parallel in a parallel series okay so I know for parallel the net resistance is how much the net resistance is one upon r1 plus one upon r2 okay so if you have a resistance like this r1 and r2 which are parallel to each other and I have applied a battery so this kind of questions you might have done before okay so the net resistance in this parallel case will be one upon r1 plus one upon r2 will be equal to one upon rnet now you have to understand that there is no direct formula for this okay we do not have any direct formula for this in the combination of errors so what I'm going to do is take a little bit of calculus which might have been taught as part of the basic module okay and then do this problem with this differentiation concept okay so if I differentiate the left hand side with respect to x okay so we have rnet one upon rnet so the differentiation for one upon x is minus one upon rnet square okay but remember this is not x and I'm differentiating with respect to x so by chain rule I will write dr net upon dx I'll write like this this will be equal to I'm differentiating now one by r1 with respect to x okay so this will be how much minus one upon r1 square dr1 by dx plus minus one upon r2 square dr2 by dx okay so what did I just do this is just basic differentiation that I did on one upon rnet one upon r1 and one upon r2 so what can I do from here I can just cancel out all the dx's because dx's are common here so dx dx dx would be cancelled out so if I write this formula the formula would look something like this dr net by rnet square will be equal to dr1 by r1 square plus dr2 by r2 square where did the minus signs go there's minus everywhere so I just cancelled out the minus signs from the left hand side and the right hand side okay so from here there's a very important statements when errors are small dr represents what dr is nothing but change very small change okay now for very small values of errors dr net is nothing but delta rnet okay so delta rnet by rnet square will be equal to delta r1 by r1 square plus delta r2 by r2 square okay again don't be afraid that okay so there's a calculus involved in this is it is it very complex the thing is the basic differentiation that we have taught in class only one formula that I have used other than that everything is same and the reason I have to use that here is because there is no direct set pattern to you know solve this kind of a problem okay we we learned the combination for z is equal to a plus b we learned the combination for you know a multiplied by b you have also done it for a raised to power m divided by b raised to power n so all these combination we have studied okay but this one is and it doesn't fit into any of these categories so that that is why I had to involve calculus here okay so anytime you get something like this that does not follow any of these combination all you have to do is a little bit of differentiation and then cancel out the dx and then just at the last point in time just say key for small values of small values and write this down for small values of error dr is nothing but it is delta r okay so whatever's in in case of very small errors delta r is nothing but dr okay so that is the concept that I use here from here from here all I have to do is plug in the values delta r net is what I need to find out r net I can find out from normal formula so if 200 ohms and 200 ohms are arranged in parallel what is going to be the r net so 1 by r net will be equal to simply 1 by 100 plus 1 by 200 okay so r net will be 100 into 200 divided by 100 plus 200 okay so that means 200 into 100 divided by 300 so to this cancels out okay so you have 200 divided by 3 200 divided by 3 okay so this would be close to how much will be this so 200 divided by by 3 3618 3618 to 66.66 okay so this is the r net so I'll take this calculation little bit away from this because I was calculating the error no so while calculating the error I will need the value for r net so I'm gonna I'm gonna use this later on for now I'm going to move this I'm gonna move this part elsewhere okay so I got the value of r net now delta r net divided by 66.66 square is going to be equal to delta r1 by what is it okay I also know the value of r delta r1 right in the question it is already given delta r1 is delta r1 is 3 ohms okay so 3 by 100 square 3 by 100 square plus this was what was the error in the second one the second register the error is 4 ohms so 4 by 200 square 4 by 200 square so from here I'll be able to figure out what is the value of delta r net okay so this will calculate when you calculate this one when you calculate this one the value comes out to be 1.8 it will come out to be 1.8 ohms okay so your final answer would be nothing but 66.66 plus minus 1.8 ohms this is the r net if you arrange the see arrange the registers in parallel condition okay any doubt in this please put it in the comments okay so this is how you do this is how you do the question okay okay let us do this question again one other ncrt question this is okay so the question says key a new system of length is chosen okay we know we know key in SI system one meter means there is a specific length that we take as one meter but this is a new system of units where one whatever this one unit of length it is different from one meter okay it is so chosen in such a way that the speed of light in vacuum is unity okay usually what is the speed of light speed of light is 3 3 into 10 raised to power 8 meters per second is the speed of light but in the new system let's call this l dash okay l dash is a new unit for length in this new new system the speed of light is one l dash per second one l dash per second okay what is this l dash l dash is they're saying that there is a new length system here new a new unit of length that i'm representing with one n dash okay so comparing these two equations you will understand okay 3 into 10 to power 8 meters is one l dash how did i get this technically these these two uh you know values may appear different okay one on the left hand side you have 3 into 10 power 8 uh is equal to one one on the right hand side but the actual speed of light is not subjected to what we choose as a unit yes yes or no because speed of light is a constant no matter what kind of system you use if you use a centimeter system if you use uh you know the si system or any imaginary system the actual speed of light is not going to change so that is why i put a equality sign in between and from there i understood oh in this new system one l dash which is the new unit of length is actually 3 into 10 to power 8 meters in the original length in the original uh the the si unit system it is actually 3 into 10 to power 8 now they're asking what is the the question is what is the distance between sun and earth in terms of unit of light takes uh 8 minutes and 20 seconds to cover this distance okay so this is another fact which is given to us key light takes 8 minutes and 20 seconds so this means 86 uh 48480 seconds plus 20 seconds so this this means 500 seconds okay so that means light takes 500 seconds from uh to reach from sun to earth so that means the distance between the sun and earth is how much speed into time yes distance is equal to speed into time which means 3 into 10 to power 8 into 500 okay so the distance is how much the distance is uh let's not even calculate let's keep it like this this is this is given to us in meters i want to know this uh you know what is this length in terms of in terms of the new unit system i know key 3 into 10 to power 8 meters 3 into 10 to power 8 meters is 1 l dash 1 l dash what i have to find out is how much is 3 into 10 to power 8 meters into 500 is in terms of l dash so the answer is very very simple cross multiply and then divide by this and then you will get this is nothing but 500 l dash okay so basic unitary method the the length between the sun and earth in the new uh new system of units is just 500 units okay so l dash is something we chose so you can write just unit here 500 units okay so that is how you do this kind of question okay let us do this question next okay it's a question based on error again so we are given equation key p is equal to a cube b square divided by okay divided by root c multiplied by d okay so this is what uh the relationship is one more few more things are given key the percentage error in a delta a by a 200 is given to us it's 1 percent okay delta b by b is given to us as uh into 100 sorry into 100 it's percentage error this is the formula i'm writing for percentage error delta b by b into 100 is given to us as 3 percent delta c by c into 100 is given to us as 4 percent and delta d by d into 100 is given to us as 2 percent okay so from combination of errors i can tell you this much k delta p by p will be actually three times delta a by a okay and plus two times delta p by b plus half times delta c by c plus delta d by d where did this come from okay so first one property i have used is key if z is a raised to power m then delta z by z is how much it is m delta a by a this is the property that i have used to figure out this this here okay and if quantities are multiplied or divided together the error between them is always added that is why i have put a plus sign between the errors of all of these okay and uh c under root c is nothing but c raised to power half that is by this half game okay that is why this half game now all i need to do is multiply by 100 on both sides so here we'll get delta p by p into 100 is equal to 3 delta a by a into 100 okay plus two times delta b by b into 100 plus half times delta c by c into 100 okay 200 plus delta d by d into 100 okay now these values are given to us these get equal one three four and two one three four and two so three into one two into what is delta b by b is three percent plus half of four percent four percent and then the last one is two percent okay so this is two percent so this will be how much this will be three plus six plus two plus two that is equal to 30 okay so the percentage error the percentage error in p is how much this 13 percent as easy as that if you have any doubts just ask me in the comment section oh there is actually one more part what what is uh how can you round off 3.763 the answer is 8 3.8 you can put it if if they are asking for the first decimal point this would be 3.8 if they're asking for the second decimal you can do it 3.76 also okay but usually they're asking usually whenever they doesn't say anything actually they're asking for the first decimal point that means here in this case it would be 3.78 why because 6 is greater than 5 once it goes beyond beyond 5 we go to the next integer so it will be 3.8 in this case okay so other than that this is pretty easy okay okay the last thing I want to tell you is that this video is designed just to let you approach the path towards ncrd it's no way the complete solution of ncrd okay this is going to help you to start solving questions from ncrd okay this will give you a basic approach to doing ncrd and again ncrd is one of the most important books okay so try to do as many questions as you can from ncrd because ncrd is one of the fundamental books for all competitive preparations as well as it will help you perform good in your school exams okay so with that I'd like to end this video and also try to do as after the video just try to do as many questions you can from ncrd you can always ask me doubts in school or classes wherever you meet me you can ask me doubts solutions to all the back questions are also easily available on the network okay so with that I will say goodbye and I'll see you again