Part B: Errors and Significant Figures

Transcript

0:00Introduction

Hello everyone! Okay, one small challenge before we start. There is a pendulum with a period of about two seconds. Your friend Rahul times one hundred oscillations, with a cheap watch that reads only to one second. Your friend Priya times just ten oscillations, with a costly stopwatch that reads to zero point one second. Who gets the smaller error in the period? Priya, obviously, right? Her watch is ten times finer! Hmm. Are you sure? Today is a problem bank session, Part B: errors and significant figures. Mixed problems, just like the exam. And look, in JEE, error questions are free marks once you know three simple tools. So, let's begin.

0:44Tool 1: percentage error from the least count

So, tool number one. You take a single reading with an instrument, just one reading. What is its error? It's simple: the least count of that instrument, okay? The instrument cannot read anything finer than that. And the percentage error is the least count, divided by the reading, times a hundred. Clear? Same least count, bigger reading, smaller percentage error. Alright, now a problem bank question. Vernier callipers, least count zero point zero one centimetre, measure five point zero zero centimetres. Screw gauge, least count zero point zero zero one millimetre, measures zero point five zero zero millimetre. The screw gauge is a hundred times finer. So its percentage error must be much smaller, right? No, no, careful! Zero point zero one by five point zero zero: zero point two percent. Zero point zero zero one by zero point five zero zero: also zero point two percent! The screw gauge measured a tiny object, so its reading is tiny too. It is the ratio that decides, never the least count alone. Got it? And a metre scale, least count zero point one centimetre, measuring twenty point zero centimetres? Zero point one by twenty: zero point five percent. So the metre scale has the largest percentage error of the three. Keep this ratio idea in mind; we will need it again very soon.

2:11Tool 2: powers multiply the percentage error

Right, tool number two: a formula with powers. Z equals A to the power p, times B to the power q, divided by C to the power r, okay? Now, C sits in the denominator. So do we subtract its error? Never! Errors always add, because we want the maximum possible error. Delta, the little triangle you see on screen, means the error in a quantity. So, delta Z by Z equals p times delta A by A, plus q times delta B by B, plus r times delta C by C, okay? Each power simply comes in front of its own relative error. With me? Quick check from the problem bank. x equals a b squared, divided by c cubed. The errors in a, b and c are one percent, one point five percent and zero point five percent. So, one times one, plus two times one point five, plus three times zero point five. That is one, plus three, plus one point five: five point five percent. See, c had the smallest error of all, but its power three tripled it. This is really important: powers are where the marks are.

3:19Tool 3: significant figures in the answer

Tool number three: significant figures in the final answer. Multiply or divide? The answer keeps as many significant figures as the factor with the fewest, okay? But add or subtract, and the rule changes. Now we count decimal places, not significant figures. The answer keeps as many decimal places as the term with the fewest. Now try this one. Two point five six into ten to the minus three, divided by one point two into ten to the power two. The calculator gives two point one three into ten to the minus five. How many figures should we keep? Yes, two! One point two has only two significant figures. So the answer is two point one into ten to the minus five. Never copy every digit your calculator shows, okay? Next, an addition. Twelve point one one, plus eighteen point zero, plus one point zero one three. The calculator says thirty one point one two three. Now tell me, which rule do we use here? Decimal places, exactly! Eighteen point zero has only one decimal place, the fewest. So we keep one decimal place: thirty one point one. That's it, done.

4:32Example 1: cheap watch or costly watch for a pendulum

Now, remember Rahul and Priya from the start? Let's settle it. There is a pendulum with a period of about two seconds. One: time one hundred oscillations, with a watch of least count one second. Two: time ten oscillations, with a watch of least count zero point one second. Which gives the smaller error in the period? Pause the video, and try it first. Done? Okay, let's check. Rahul times one hundred oscillations of about two seconds each: about two hundred seconds in total. His watch can be off by one second, its least count. So, one by two hundred: zero point five percent. Now Priya. Her watch is ten times finer, so her error must be ten times smaller, right? No, no, careful! Remember tool one: the ratio decides. She times only ten oscillations: about twenty seconds in total. Zero point one by twenty: again, zero point five percent! And the period is the total time, divided by the number of oscillations. That count is exact, with no error, so the period carries the same zero point five percent in both cases. Equally good! In fact, Rahul's cheap watch did just as well, because he timed ten times longer. Mystery solved, right?

5:53Example 2: g from free fall with its error

Now a slightly harder one, JEE level. Here, a body falls two point zero zero metres from rest, plus or minus zero point zero one, in zero point six four seconds, plus or minus zero point zero one. Find g, with its error. Pause and try. Hint: for a fall from rest, g equals two h by t squared. Done? Okay, let's check. So, g equals two h by t squared: two times two, divided by zero point six four squared. That is nine point seven seven metres per second squared. Now the errors. In the height: zero point zero one by two, that is zero point five percent. In the time: zero point zero one by zero point six four, about one point five six percent. So the error in g is zero point five plus one point five six, right? No, no, careful! This is the JEE trap. Time is squared in the formula, so its error counts two times. Delta g by g equals delta h by h, plus two times delta t by t. And the two on top? An exact number, so it carries no error at all. Clear? So, zero point five, plus three point one two. About three point six percent. Now the absolute error. Nine point seven seven times three point six percent: about zero point three five. We quote an error to one significant figure, so that becomes zero point four. The error sits in the first decimal place, so g also stops there: nine point eight, plus or minus zero point four metres per second squared. And look: the height gave only zero point five percent, but the timing gave more than three percent. So, to do better, improve the timing. Topper's habit: see which term gives the biggest share, its percentage error times its power, and measure that quantity most carefully. Got it?

7:53Quick revision: least count error, powers, sig figs

So, quick revision. One: the error of a reading is the least count, and its ratio to the reading decides the percentage error. Two: in a formula, multiply each percentage error by its power, and always add. Three: multiply or divide, keep the fewest significant figures; add or subtract, keep the fewest decimal places. Now go and try the five practice problems on this page. You will find them easy, I promise. All the best!