Hi guys, welcome back to another video. Today we are going to be revising some of the fundamentals of physics, in other words, uncertainties in experiments. By the end of this online lesson, we are definitely going to be extremely certain about the uncertainties. There are two types of uncertainties that we need to be aware of. First off, absolute uncertainties, those are the smallest measurements that can be taken.
but an instrument. Some exam boards assume that they're half of the smallest measurements, so that's why I've put both statements in. For this purpose, for this video, I'm going to assume that they're the smallest measurement that you can take, but an instrument. For instance, if I had a millimeter ruler, the smallest measurement that I'm going to take, that I'll be able to take, is a millimeter, so this will be the absolute uncertainty in my measurements. In digital instruments, however, the last digit always indicates the absolute uncertainty.
For instance, if I had a voltmeter with the following reading, 5.46 volts, I cannot really trust the last digit, as it can be 5.47 or 5.45. So in this case, we are confident that the real value is sandwiched between 5.45 volts and 5.47 volts. We can also calculate the percentage uncertainty. Our formula for the percentage uncertainty is that it equals plus or minus our absolute uncertainty divided by our experimental value, and all of this is multiplied by 100. In other words, we take the percentage of this value with respect to this value and then we, well, we times by 100. In the case of this measurement here, our percentage uncertainty will equal plus or minus our absolute uncertainty, which is 0.01 divided by our value, which is 5.46 times 100. We can put this into a scientific calculator. What we're going to get is 0.01 divided by 5.46 times 100 and this will be equal to 0.18%.
This means that we can also write our original value which was 5.46 volts as simply 5.46 volts plus or minus 0.18. If we're given the percentage uncertainty, we could really easily find the absolute uncertainty. For instance, if we had a different reading which was, let's say, 50 volts, and let's say that we're confident within plus or minus 5%, this is our reading with the percentage uncertainty, but our absolute uncertainty will just be 5% of 50. volts, which is just 2.5 volts. So the absolute uncertainty in this case will be 50 volts plus or minus two and a half volts.
Once again, this is our percentage uncertainty, and this over here is our absolute uncertainty. When solving problems on combining uncertainties, the first question that we need to ask ourselves is, what are we doing with the quantities? Are we adding them or taking them away?
Are we multiplying or dividing? Or are we raising to a power? In some problems, we might be doing all three of them.
Let's have a look at an example in each scenario. Let's focus on the first case. So if we are adding or taking away quantities, we need to add the absolute uncertainties. For instance, we measure two voltages, 5.0 plus or minus 0.1 volts, one with a digital voltmeter, and another measurement with a different voltmeter, which gives us 4.0 volts plus or minus 0.2 volts across some different component. Find the absolute and the percentage uncertainties in the total voltage.
Now because to find a total voltage we need to add the two voltages, So let's just write down over here that V total will be equal to 5.0 plus 4.0 which is 9.0 volts. So in this case we are adding uncertainties and we're adding the absolute uncertainties. The absolute uncertainty in this measurement will simply be the algebraic sum of the absolute uncertainty. So 0.1 plus 0.2 which is 0.3.
So our absolute uncertainty in this measurement, let's write it with the measurement itself, will be 9.0 volts plus or minus 0.3 volts. And in this case, this over here is our absolute uncertainty. We can also find the percentages for uncertainty, of course, simply by using the formula that percentage uncertainty is plus or minus the absolute uncertainty times 100 divided by the value. So in this case our absolute uncertainty is plus or minus 0.3. We're going to divide this by our value for the total voltage which is 9.0.
We're going to multiply this by 100 and that's going to give us about 3.3 percent. On the other hand, if we are multiplying or dividing quantities, we need to add the percentage uncertainties. For instance, we can have a look at this example. They're asking us to find the percentage uncertainty in the resistance of a component if the reading for the voltage is 5.0 volts plus or minus 0.1 volts, and the reading for the current is 1.1 amp plus or minus 0.1 amp. Remember, R is V over I.
So in this case, we are dividing quantities, so we need to add the percentage uncertainties. The way I'm going to write this is that the percentage uncertainty in R, I'm just going to write PU for short, will be equal to the percentage uncertainty in V, plus... the percentage uncertainty in I, like so.
So the percentage uncertainty in the resistance will be equal to plus or minus 0. divided by 5.0 times 100 plus the percentage uncertainty in I, which is plus or minus 0.1. divided by 1.1 times 100. And if we calculate this out, we are going to get about 11% up to two significant figures, which is our final answer. Finally, if we are raising a quantity to a power, let's say that the quantity is y, could be anything, it could be volume, it could be length, it could be time, voltage, etc.
The percentage uncertainty in that quantity raised to the power of n is equal to n multiplied by the percentage uncertainty in that quantity. Let me give you an example. Find the percentage uncertainty in the volume of a cube of side 4.0 plus or minus 0.1.
meters. First off, the volume is found by cubing 4. So this will be equal to 4 raised to the power of 3, which equals 64 cubic meters. Now the percentage uncertainty in the volume will actually be equal to 3 times the percentage of this measurement because we have raised the measurement to the power of 3. So this will be equal to 3 times the percentage in certainty. This measurement will be 0.1 divided by 4.0 times 100. And if we put that in a calculator we are going to get 7.5 percent.
So in fact our measurement can be written or our calculation for the volume can be written as 64 cubic meters plus or minus seven and a half percent. So just to summarize guys, if we are adding or taking away quantities, we add the absolute uncertainties. If we multiply or divide quantities, we add the percentage uncertainties.
If we raise a quantity to a power power, we multiply the percentage uncertainty in that quantity by the power itself. Okay folks, well this was most of uncertainties. There's only one thing remaining for you to revise, and that is how to find the uncertainty from a graph. I'm including a link in the description of this video so that you guys could carry on with your revision.
Thank you very much for watching, I hope this was useful and if there are any questions please feel free to drop a comment and I'll see you in the next video.