Video Transcript
Find the difference in the percent uncertainties of the two following measurements: 10 plus or minus 0.5 seconds and five plus or minus 0.1 seconds.
To find what we’re looking for, we will need to know how to calculate the percent uncertainty of a measured value. When we express a measured value that has uncertainty, we usually write it in the form 𝑎, the measured value, plus or minus 𝑏, which is the uncertainty in the measurement. And since this is the value of some physical quantity, we also include appropriate units.
Now, when we write the value this way, the uncertainty is written as what we call the absolute uncertainty. That is, it’s just a number and doesn’t depend on the measured value. The percent uncertainty, which is what we are looking for, expresses this absolute uncertainty as a fraction of the measured value. As a mathematical formula useful for calculating, we can write that the percent uncertainty in a measured value is 𝑏 divided by 𝑎, where 𝑏 is the uncertainty and 𝑎 is the measured value, times 100 to turn that fraction 𝑏 divided by 𝑎 into a percent. Note that the percent uncertainty is a unitless quantity because both 𝑏 and 𝑎 have the same units. So 𝑏 divided by 𝑎 is unitless.
Now, we are given two measured values, and both of them are expressed using absolute uncertainties like we first talked about. For 10 plus or minus 0.5 seconds, 0.5 is the absolute uncertainty, what we called 𝑏, and 10 is the measured value, what we called 𝑎. Substituting for 𝑎 and 𝑏 in our formula for percent uncertainty, we have 0.5 divided by 10 times 100, which is equal to five percent. This actually gives us another way to express our original value. Instead of writing 10 plus or minus 0.5 seconds, we can write 10 seconds plus or minus five percent.
Let’s now apply the same steps to five plus or minus 0.1 seconds. 0.1 is the uncertainty, and five is the measured value. Dividing the uncertainty by the measured value and multiplying by 100 gives us 0.1 divided by five times 100, which is two percent. And as before, this lets us rewrite five plus or minus 0.1 seconds as five seconds plus or minus two percent.
The final value we’re looking for is the difference between these two percent uncertainties. Five percent minus two percent is three percent, and this is the value that we are looking for.
