A block is initially at rest on a frictionless tabletop. One end is attached to a wall, as shown in the figure. The block is pulled away from the wall. Use the graph shown to answer the following question.
Before we get to that question, let’s take a look at this figure that shows the block attached to a wall by a spring. Here’s our block. Here’s our spring and here’s the wall the spring is attached to. Starting from this position, We’re told that the block is pulled away from the wall to the right as we’ve drawn it. Knowing all this, let’s move on now to our question.
Which graph correctly represents the magnitude of the force exerted on the block by the spring as a function of the displacement of the block from its initial position?
We see we have five separate graphs a, b, c, d, and e, which purport to indicate the magnitude of the force exerted on the block by the spring as a function of the displacement of the block from its initial position. We can see that the axes for all of these graphs are unlabeled. But we’re told in this question what they’re meant to indicate. The vertical axis is meant to indicate the magnitude of the force exerted on the block by the spring. We can represent that force using the letter capital 𝐹. And so we will sketch that in on all our graphs, just to remind us of what these graphs are meant to indicate.
That force magnitude is being plotted as a function of the displacement of the block from its initial position. And that tells us what the horizontal axis of each of these plots indicates. Looking back at our sketch of the block and the spring, if we say that this is the initial position of the block, then any deviation from that position can be labeled the displacement 𝑋 of the block. And it’s that displacement that the horizontal axis of each of our five graphs is meant to indicate. So each graph is meant to show us a plot of the force on the block as a function of the displacement of the block from its initial equilibrium position. And of course, what we want to solve for is which of the five graphs shows us the correct relationship between force and displacement. To start figuring that out, let’s clear a bit of space on screen.
Whenever we’re talking about masses attached to springs and displacement of those masses, it can remind us of a law known as Hooke’s law. This law tells us that there is a particular relationship between the force that a stretched spring exerts on a mass and the displacement of that mass from its equilibrium position. We can state the law this way. It says that that force, the force exerted by the spring on the mass, is directly proportional to, that’s what that symbol means, the displacement of the mass from its equilibrium position. Another way to say this is the force is proportional to the stretch. Now, right away, we can use Hooke’s law to eliminate one of our answer options from contention.
Say that our mass is at its initial position. In other words, its displacement from that position is zero. Well, Hooke’s law tells us that the force then exerted by the spring on the mass at that point is also zero, zero force for zero displacement. That means that any of our answer options would show a line that does not pass through the origin, the point where the force and displacement are both zero. Then, according to Hooke’s law, that can’t be an accurate representation of this relationship between force and displacement. We see that option a is like that. It shows that, for zero displacement, there’s a nonzero force. Hooke’s law says this can’t be the case. Let’s keep going with this law to find out more.
If force is proportional to displacement, then that means that if I double the displacement of my mass, then I also double the force involved. And this also means if I triple the displacement, going up to three times some displacement 𝑋, then I triple the force that the spring applies to the mass. What we’re seeing is a linear relationship between force and displacement. And this means whatever our correct answer is, it must show this linear relationship. If I double the force, I must double the displacement and vice versa. Let’s take a look, for example, at the graph marked out c.
This graph claims that there is a sinusoidal relationship between force and displacement. When we first pulled the block away from its initial position, the force on it increases. But then, as we continue to pull it away, the force decreases and eventually becomes negative. But this goes against the linear relationship we see in Hooke’s law. So this graph, choice c, isn’t our answer. Let’s move on now to consider graphs d and e. For each of these curves, we see the curve is not a straight line. That also goes against this linear relationship we’re discovering as we look at Hooke’s law. So these graphs don’t represent the correct relationship between force and displacement either.
Finally, looking at graph b, we see that this line passes through the origin, which it must to agree with Hooke’s law. And that also, given a certain force and displacement — we can say this amount of force and this amount of displacement — if we double either one of those parameters by going up to this point, then we also double the other parameter. In other words, we are seeing this linear relationship that Hooke’s law indicates. So it’s graph b, which correctly represents the magnitude of the force exerted on the block by the spring as a function of the displacement of the block from its initial position.