Errors in the Final Result

Transcript

0:00Introduction

Hello everyone, here is a small puzzle before we start. You have a cube, and you measure its side and its mass. Each reading is good to about half a per cent, which sounds excellent, right? Now you work out the density. So how good is the density, half a per cent, or maybe one per cent? Hold on to your guess, because the real answer may surprise you. Questions on errors in a final result come up often in JEE, so let's get this right.

0:32One reading, then several readings

So, let us start with just one reading on an instrument. That single reading is uncertain by the least count, okay? Now suppose you take several readings. Then you report the mean, the average of all of them. Clear? Each reading differs from the mean by a small amount, called its deviation. Let me ask you this: keep the plus and minus signs, and add all the deviations. What do you get? You get zero, which would mean no error at all, and that cannot be right. So we take the magnitudes only, which means we ignore the signs, and then find their mean. This is the mean absolute error, right? We round it to the least count, and we write the result as the mean, plus or minus this error. This plus or minus sign shows the error on both sides of the value. Got it?

1:26Sum or difference: absolute errors add

Now let us combine two measurements. Here are two lengths: twelve point four, plus or minus zero point one, centimetres, and ten point two, plus or minus zero point one, centimetres. Their difference is two point two centimetres, but what is its error? Is it zero point one minus zero point one, which is zero? That is the trap, so be careful. One reading can be too high while the other is too low, and we always allow for that worst case, so here errors never cancel. For a sum or a difference, the absolute errors add. Delta of A plus or minus B equals delta A plus delta B, okay? Delta, the Greek letter shaped like a little triangle, means the error in a quantity. So here, zero point one plus zero point one gives zero point two. The difference is two point two, plus or minus zero point two, centimetres. Clear? So the per cent error is zero point two over two point two, times one hundred, which is about nine per cent. Those two wavy lines mean approximately, okay? Each length alone was known to better than one per cent! So remember this: when two quantities are nearly equal, their difference is poorly known. Right?

2:40Product, quotient, powers: relative errors add

Alright, now products, quotients and powers. Let capital Z equal capital A to the power small a, times capital B to the power small b, over capital C to the power small c. Now tell me, capital C sits in the denominator, so should its error be added or subtracted? It is added, because here too we allow for the worst case, so errors never cancel. So delta Z over Z equals small a times delta A over A, plus small b times delta B over B, plus small c times delta C over C, okay? Each fraction like delta A over A is called a relative error. So the relative errors add, with the powers as weights. And when you multiply a relative error by one hundred, you get the per cent error. With me? The quantity raised to the highest power, and the one with the smallest value, need the most careful measurement. Clear? And one last rule for writing the answer. The result is rounded so that its last figure is the first uncertain one. Got it?

3:43Why we time many oscillations

Now a case from the pendulum. Time n oscillations, and call the total time small t. Then the period, capital T, is small t over n, okay? Suppose twenty oscillations take thirty six point zero seconds, on a watch of least count zero point two seconds. So the period is one point eight zero seconds. Here is my question: is the error in the period still zero point two seconds? It is much smaller, because that zero point two is the error in the whole time, shared over all twenty oscillations. So delta capital T equals delta small t over n, and zero point two over twenty gives zero point zero one seconds. The period is one point eight zero, plus or minus zero point zero one, seconds. Right? Now divide delta capital T by capital T, and the n cancels. So delta capital T over capital T equals delta small t over small t, the same relative error, okay? And small t grows when you count more oscillations, while the least count stays the same. That is why we time many oscillations.

4:52Example 1: percentage error in the volume of a sphere

It is your turn now, with an easier one. The diameter of a sphere is one point two six centimetres, by a callipers of least count zero point zero one centimetres. Find the percentage error in its volume. Pause the video, and try it first. Done? Okay, let's check. The volume, capital V, is pi over six times d cubed, where d is the diameter. Pi over six is just a number with no error, and the diameter comes with a power of three, right? It is a single reading, so delta d, its error, is simply the least count, zero point zero one centimetres. Clear? So the per cent error is three times zero point zero one over one point two six, times one hundred, which is about two point four per cent. Forget that three, and you lose the mark, so please be careful.

5:43Example 2: density of a cube with its error

Now let us finish our cube from the start, which is a little harder. The side is two point zero zero, plus or minus zero point zero one, centimetres, and the mass is twenty one point six, plus or minus zero point one, grams. Find its density with the error. Pause the video, and try it first. Done? Okay, let's check. Density, the Greek letter rho, is m over s cubed, where m is the mass and s is the side. The side cubed is eight point zero zero, so twenty one point six over eight point zero zero gives two point seven zero. The unit is grams per cubic centimetre, which is what that little minus three on the centimetre means, okay? Now for the error. The mass is on top, and the side is at the bottom with a power of three. So how do the two relative errors combine? Yes, exactly, they add, with the power as the weight. So delta rho over rho equals delta m over m, plus three times delta s over s, right? Now the numbers. Zero point one over twenty one point six, plus three times zero point zero one over two point zero zero, gives zero point zero two zero, which is two per cent. So there is our surprise. Mass and side were each good to about half a per cent, yet the density is at two per cent, because the side counted three times, okay? Now we need the absolute error. Delta rho is zero point zero two zero times two point seven zero, which is about zero point zero five. So the density is two point seven zero, plus or minus zero point zero five, grams per cubic centimetre. Clear? And see, the error sits in the second decimal place, so that last zero in two point seven zero is the first uncertain figure. That is our rounding rule in action.

7:30Quick revision: how errors combine and how to round

So, quick revision. For a sum or a difference, absolute errors add, they never cancel. For products, quotients and powers, relative errors add, with the powers as weights. And round the result so that its last figure is the first uncertain one. Now go and try the practice problems on this page. All the best!