Question Video: Finding the Uncertainty in a Given Quantity Squared Physics • First Year of Secondary School

Video Transcript

In an experiment, a quantity 𝑞 is found to have a value of 15 plus or minus 0.3. What is the percent uncertainty in 𝑞 squared?

We’re told here the value of this quantity 𝑞, which we see involves an uncertainty. In order to solve for the percent uncertainty in 𝑞 squared, it will first be helpful to solve simply for the uncertainty in 𝑞 squared. If we write out 𝑞 times 𝑞, it looks like this. And note that we’re effectively taking one number with an uncertainty and multiplying it by another number also with an uncertainty. If we imagine a general case where we have a number 𝑎 with an uncertainty 𝜎 𝑎 and another number 𝑏 with an uncertainty 𝜎 𝑏 and if we say further that a third number 𝑐 is equal to the product of 𝑎 and 𝑏, then we can write that the uncertainty in this product is equal to 𝑏 times the uncertainty in 𝑎 plus 𝑎 times the uncertainty in 𝑏.

Applying this relationship to our scenario, we can say that this first 15 is 𝑎 and the first 0.3 is the uncertainty in 𝑎, 𝜎𝑎. The next 15 we can think of as 𝑏, and the next 0.3 as the uncertainty in 𝑏. When we find this product 𝑞 squared then, that’s equal to 𝑎 times 𝑏, 15 times 15, that’s 225, plus or minus some uncertainty. According to our equation here, that uncertainty equals 𝑏 times the uncertainty in 𝑎 plus 𝑎 times the uncertainty in 𝑏. It so happens since we’re squaring 𝑞 that 𝑎 and 𝑏 are equal and so are the uncertainties in these two values. If we multiply 15 by 0.3, that gives us 4.5 and the same thing holds true for this product. Our total uncertainty then is 4.5 plus 4.5, or nine.

Now, if we just wanted to find the uncertainty in 𝑞 squared, our job would be done. But our final answer will be the percent uncertainty in 𝑞 squared. To understand how to express this uncertainty as a percent uncertainty of this number, let’s imagine we had some round value, say 100, that had an uncertainty of exactly one. We know that one is one percent of 100. And this shows us that if we want to solve in general for the percent uncertainty of some value we’ll call 𝑎, then that will be equal to the uncertainty in 𝑎 divided by 𝑎 times 100 percent. Note that if we put one in place of 𝜎 sub 𝑎 and 100 in place of 𝑎, then we do indeed get one percent.

This shows us that the percent uncertainty in 𝑞 squared is nine divided by 225 multiplied by 100 percent. When we calculate this expression, we get exactly four percent. This then is the percent uncertainty in 𝑞 squared.