Video: Rearranging Formulas for Physical Quantities

In this lesson, we will learn how to change the subject of a formula and recognise valid rearrangements of combinations of formulas.

15:07

Video Transcript

In this video, we’re talking about rearranging formulas for physical quantities. Now, we can see in this picture a formula that needs to be rearranged. But that’s because it’s written incorrectly. What we’ll talk about in this lesson, though, is different from that. It’s not rearranging incorrect formulas to make them correct. But instead, we’ll take formulas that are already physically accurate. And we’ll learn how to express them in different ways. We can start out with the correct expression for Newton’s second law of motion. The net force on an object is equal to that object’s mass times its acceleration. Written this way, this formula has a subject. That subject is the force 𝐹. The way we can recognise a subject in an equation is it’s the term which is on one side of the equation all by itself. Right now, we have this single term, the force 𝐹, followed by an equal sign, which means that 𝐹 is the subject of this equation.

But let’s say that we were given an example exercise, something like this. Say that we were pushing a glass across a smooth tabletop and that we were applying a known constant force to the glass with a given mass. If we wanted to solve for the acceleration of the glass, we could use Newton’s second law to do that, but not in the form it’s currently in. Since we already know the force 𝐹, having that be the subject of our equation is not what we want. Instead, we want the acceleration, 𝑎, the variable we’re trying to solve for, to be the subject. To make that happen, it will be necessary to rearrange this equation. Whenever we want to rearrange a formula, that requires some algebraic operations, adding, subtracting, multiplying, dividing.

One way we can approach this process is to look at the formula in its current given form and recognise what we want the subject of the new rearranged formula to be. As we’ve seen in our case, that’s the acceleration 𝑎. Now, if 𝑎 is going to be the new subject, that means that this variable needs to be all by itself on one side of the equation. This means that everything on the right-hand side of this equal sign besides the acceleration 𝑎 needs to be moved to the left-hand side. If we do that, then 𝑎 will be all by itself on one side of the equation. It will be the subject. In this instance, we can see that, to make 𝑎 the subject of our formula, all we need to do is move the mass 𝑚 from the right side to the left side. But then, that raises the question, how exactly do we do that? Do we add 𝑚, subtract it, multiply it, divide it? Those are our four options. And to figure out which one to pick, we can ask ourselves this question. What mathematical operation could we perform on the right-hand side of this equation so that 𝑚 disappeared from that side?

The reason we’re asking that question is because if 𝑚 did disappear, that would leave 𝑎 all by itself. It would be our formula’s new subject. Well, as we think about that, what if we divided the right-hand side of this equation by 𝑚. If we did that, then we would have one factor of 𝑚 in the numerator and one factor in the denominator. They would cancel one another out. That’s good. That’s what we wanted to have happen. But we need to be careful. Since we’re working with an equation, that means whatever is on the left of this sign should be equal to whatever is on the right. And that means that if we do something to one of these two sides and in this case we divided that side by 𝑚. Then in order to maintain this equality, we need to do the exact same thing to the other side of the formula. Since we divided by 𝑚 on the right-hand side, let’s do that on the left-hand side as well.

If we clear away some of this clutter, here’s what we’ve done to this 𝐹 equals 𝑚𝑎 equation. We’ve divided both sides of the equation by the mass 𝑚. And as we saw, that means this term cancels out on the right-hand side. Getting rid of that cancelled term, we can now see the new rearranged form of our formula. Just as it’s accurate to say that 𝐹 is equal to 𝑚 times 𝑎, we can also equivalently say that 𝐹 divided by 𝑚 is equal to 𝑎. And we’ve now rearranged our formula so that our new subject, the acceleration 𝑎, is the variable we want to solve for. And from here, all we need to do is substitute in the given values of 𝐹 and 𝑚, divide 𝐹 by 𝑚. And that will equal 𝑎. Let’s take a moment to consider the steps that we took in order to convert our original formula into a more useful rearranged form.

Even though we were working with a specific formula, Newton’s second law of motion, the steps we followed, the process applies in general to rearranging equations. Step one was we identified what we wanted to be the subject of the formula. The way it was given to us. That subject was the Force 𝐹. But we wanted it to be the acceleration 𝑎. And then, our second step was to add, subtract, multiply, divide any and all of those in order to isolate our new subject. And when we say isolate, we mean put that variable on one side of the equation all by itself, just like we did with the acceleration 𝑎. Now, we might wonder, how will we know which of these four operations or some combination of them should we apply?

The specific approach we take will depend on the formula we’re working with. But here’s a general rule of thumb. If there’s a variable in our formula that we want to move through rearranging. In the case of 𝐹 equals 𝑚𝑎, that variable was the mass 𝑚. Then, however, that variable combines with the variable that we want to be the subject. We pick the opposite mathematical operation by which they’re combined in order to isolate our subject. Now that may sound complicated. But all it means is that if the variable that we want to move is multiplying the variable that we want to be the subject, then we should divide by the variable we want to move. And the opposite is true too. Say that we somehow had an equation where on the one side we had 𝑎 divided by 𝑚. If we wanted to isolate 𝑎. If we wanted that to be the subject of this equation, then since 𝑚 currently is dividing into 𝑎, we would multiply this side, as well as the other side of the equation, by 𝑚. That would be a step toward isolating the subject that we want.

So if we have an equation where our subject is 𝑥 and we’re adding something to it. Then this rule of thumb says to subtract that from both sides or if the operation was different. Say instead of 𝐴 being added to 𝑥, 𝐴 was being subtracted from it. Then we would pick the operation opposite of subtraction, that’s addition, and add 𝐴 to both sides of the equation. Or, as we’ve seen, if 𝐴 multiplies, what we want to be the subject, then we divide both sides by 𝐴 and so on. That’s what we mean when we say that if a variable is mathematically combined with a variable that we want to be our subject, then combining that variable that we want to rearrange in the opposite way will help us do that. So this is our two-step process for rearranging equations to solve for a variable of interest.

Now there’s another kind of rearrangement we’ll want to be aware of. In this scenario, rather than just having one equation, we have two. And the rearranging we want to do involves a substitution. Thinking back to our example of the sliding glass, let’s say that, along with knowing the force applied to it and the mass of the glass, we also know how much the velocity of the glass changed, we can call that Δ𝑣, over some time interval. Knowing all this, what if we wanted to solve for that time interval? And we can call it Δ𝑡. Based on the two equations we have, 𝐹 is equal to 𝑚𝑎. And 𝑎 is equal to Δ𝑣 divided by Δ𝑡. We can see that neither one of these equations by itself will let us solve for Δ𝑡. But if we combine them, then this becomes possible.

Considering these two formulas, notice that the acceleration 𝑎 appears in both of them. And what’s more, in the equation on the right, the acceleration is the subject of that formula. That means that everything on the right-hand side of this equation is equal to 𝑎. And now, this is where the substitution step comes in. Since this 𝑎 here and this 𝑎 here are exactly the same, then if some other expression is equal to 𝑎, as we know Δ𝑣 divided by Δ𝑡 is. Then that means we can substitute that expression for 𝑎 into this expression of 𝑎 in our 𝐹 equals 𝑚𝑎 equation. When we do that, our new combined equation looks like this. 𝐹 is equal to 𝑚 times Δ𝑣 divided by Δ𝑡. We can now remember that it’s Δ𝑡 that we want to solve for. We want that to be the subject of this formula and that from our problem statement, we’re given the force 𝐹, the mass 𝑚, and the change in velocity Δ𝑣.

So now we can follow our process for rearranging an equation to change the subject of that formula. We want the subject in this case to be Δ𝑡. And we know that means having Δ𝑡 in the numerator. Right now, of course, it’s in the denominator. But the way to change that is to multiply both sides of this equation by Δ𝑡. When we do that, looking on the right-hand side, we see that Δ𝑡 in the denominator cancels with the one upstairs. But now, on the left-hand side, we have Δ𝑡 times 𝐹.

Since we want Δ𝑡 all by itself, we need to think of some operation to perform to move 𝐹 to the other side of the equation. Since 𝐹 is currently multiplying Δ𝑡, our first idea is to divide both sides of the equation by 𝐹. In this way, we’ll cancel that term out from the left-hand side. And then, once we clear that cancelled term away, notice that we now have a rearranged formula, where Δ𝑡 indeed is the subject. If we were going to calculate Δ𝑡, our next step would be to substitute in the given values for the other variables. To get this result, we use substitution to combine two separate equations and then rearrange the resulting equation to solve for the variable of interest.

Now, before we get to an example exercise, there’s one practical matter to consider as we talk about rearranging formulas. And that is that sometimes this rearranging is just mathematically confusing. For example, say that we want to solve for a value, we’ll call 𝑥. The only trouble is 𝑥 is equal to a complex expression. In fact, this is literally a complex fraction. See that we have a fraction in the numerator and a fraction in the denominator. How can we simplify this expression? Well, let’s say that when we get a final expression for 𝑥, that we’re okay with a simple fraction, but not a complex one. In other words, in our final expression for 𝑥, we should only see one divided by sign, whereas now we have three. So we’ll want to rearrange this expression on the right to get rid of some of these divisions.

One way we can think of doing this is to try to make the denominator of this complex fraction equal to one. If we did that, then our fraction overall would be 𝑎 divided by 𝑏 divided by one. But we know that any number divided by one is just the number itself. So if we could get this denominator to equal one, then we could have a simplified expression for our variable 𝑥. Now, the quickest way to turn some value into one is to multiply that value by its inverse or its reciprocal. The inverse of 𝑐 divided by 𝑑, where 𝑐 and 𝑑 are just variables, is 𝑑 divided by 𝑐. And notice that if we multiply these fractions together, the 𝑐 in the numerator will cancel with the 𝑐 in the denominator. And the same thing will happen for the 𝑑. In other words, this whole expression will simplify to one. But wait a moment. We’ve just multiplied the denominator of our fraction by some number. Yet this denominator and the fraction it’s a part of is part of an equation, where one side is equal to the other. This means if we do one thing to one side of the equation, we have to do the same thing to the other side, to keep the equality true. Or there’s an alternative.

Another way that we could maintain this equality, expressed in this formula, is to multiply the numerator of the right-hand side by the same fraction we divided by. In other words, to multiply the numerator by 𝑑 divided by 𝑐. The reason this keeps the equality true is that we’re multiplying by the same number that we’re dividing by. And any number divided by itself is equal to one. So we’re essentially multiplying the right-hand side of this original equation by one. That’s what it means to multiply 𝑑 over 𝑐 divided by 𝑑 over 𝑐. So we’re effectively multiplying by one. But in doing this, the values in our denominator all cancel out an equal one. What we’re left with then is 𝑎 divided by 𝑏 times 𝑑 divided by 𝑐 all over one. But as we said, any number divided by one is equal to the number itself. So our final expression simplifies to 𝑎 times 𝑑 divided by 𝑏 times 𝑐. This is a simple fraction, which is equal mathematically to the original complex fraction we started with. When we rearrange equations, in general, it’s helpful to replace complex fractions with simple ones, like we’ve done here.

Now that we’ve seen a few different approaches for rearranging formulas, let’s get a bit of practice with these ideas through an example.

The speed of an object is equal to the distance the object has traveled divided by the travel time taken, expressed as speed equals distance divided by time. What is the formula for the object’s distance traveled?

Okay, in this exercise, we’re given this relationship between speed, distance, and time. Written out this way, the subject of this equation is the object’s speed. That’s the term that’s by itself on one side of the formula. What we want to do is rearrange this equation so that distance, rather than speed, is the subject. Now that we know what the new subject of our rearranged equation will be. We want to figure out what operations we’ll need to perform, addition, subtraction, multiplication, division so that this term, distance, is by itself on one side of the formula. We can see right away that the only other term on the right-hand side of our equation, along with distance, is time. This, then, is the only term we’ll need to rearrange so that distance becomes our subject. Since right now, time is divided into distance, in order to move it from the right-hand side to the left-hand side, we’ll multiply both sides by that same term.

There are two reasons that we take this step. For one thing, multiplying both sides by time will cancel that term out on the right-hand side. But then, we needed to multiply both sides, and not just one, by this variable, so that our equation would hold true. If we do something to one side of the equation, we need to do the same thing to the other side. That’s why we’ve multiplied the left-hand side by time as well. With the factors of time in the right-hand side cancelling out, now the only thing on that side of the equation is our subject, distance. And we see that distance is equal to time multiplied by speed. This is the rearranged form of our formula, where distance, rather than speed, is the subject.

Let’s summarise now what we’ve learned about rearranging formulas for physical quantities. Starting off, we saw that, in a formula, for example, an imaginary formula 𝐴 is equal to 𝐵 times 𝐶, the quantity that’s isolated on one side of that formula is called the subject. In the case of this equation, 𝐴 equals 𝐵 times 𝐶, 𝐴 is that subject. We then saw that rearranging an equation to change the subject of the equation involves a two-step process. First, we identify the new subject. That’s the variable that we’ll now write the equation in terms of. Then second, we perform mathematical operations, addition, subtraction, multiplication, and division, as needed, in order to isolate that new subject. And lastly, we saw that approaches called substitution and simplifying complex fractions are also rearrangement techniques.

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