Force & Potential Energy - Online Tutor, Practice Problems & Exam Prep

Potential energy graphs illustrate the relationship between potential energy and position, allowing for the analysis of mechanical energy conservation. The total mechanical energy is the sum of kinetic energy (KE) and potential energy (PE), expressed as $E=K+U$. The slope of the graph indicates the force's direction: a negative slope corresponds to a positive force, while a flat slope indicates equilibrium. Stable equilibrium occurs at potential energy minima, while unstable equilibrium occurs at maxima, affecting an object's return behavior when displaced.

Potential Energy Graphs

Guys, occasionally in some problems, you have to calculate things like speeds and energies, but instead of using your energy conservation equations, you're going to have to use these things called potential energy graphs. I'm going to show you how to use and sort of read these potential energy graphs. We're going to work out this problem together. Let's check this out. The whole idea here is that potential energy graphs will graph the potential energy of an object on the y-axis versus the position of that object on the x-axis. These can actually tell us some pretty interesting things about the motion of an object without having to get into your crazy work energy equations. Let's go ahead and get to the problem here. We have a marble that is following this potential energy graph, and we're going to release the marble from rest at x equals 1. I'm going to locate x equals 1 on my diagram, figure out where that point is. I'm going to call this point A, and we're releasing this marble right here with a speed of 0. In this first part, we want to calculate the total mechanical energy of the marble. We know exactly how to do that. The mechanical energy, right at point A, is just going to be k + u. So that's just going to be the kinetic at A plus the potential at A. So how do we calculate this? Well, remember kinetic is always related to speed, and we just said that the speed at A is going to be 0. So there actually is no kinetic energy here at point A. So all of the mechanical energy is really just going to be the potential energy. Now, what I want to point out here is that this potential energy isn't necessarily gravitational potential. It's not spring potential energy. It's kind of just some vague potential energy that I don't really know what it's from. So we actually don't use an equation to calculate this. We're just going to look at this. We're just going to look at the value on the graph at point A to figure this out. So here at point A, the y-value is equal to 8 joules, and that's our potential energy. So, really, our total mechanical energy is actually equal to 8 joules. That's the answer.

So what ends up happening is that in general, the mechanical energy at any point is going to be the k + u at that point. Now you only have to solve for this once in a problem because we know that mechanical energy is always going to be conserved if the work done by non-conservative forces is 0, and that's always going to be the case in these problems. So what I like to do is I like to draw a little horizontal line once I've calculated the mechanical energy, and I say that this is the mechanical energy of this object throughout the entire motion. It always has to equal 8 joules. This marble, no matter where it is on the graph, always has to have 8 joules of mechanical energy. Let's take a look at part b now. Part b, we're asked to calculate the kinetic energy at x equals 3. So x equals 3 is right over here, so I'm going to call this part, point B. And before we actually go ahead and calculate this, I kind of want to use a roller coaster analogy. I like to think of this marble as kind of like a roller coaster cart that's traveling on some tracks. The potential energy graph is basically the roller coaster track. So as we're going from A to B, we're going to be going downhill, and therefore, we're going to gain some speed and therefore some kinetic energy. And that's what I want to figure out. So how do I figure out k_B? Well, remember, the whole idea with these problems is that the mechanical energy is conserved, so that I can say that the mechanical energy at B is just going to be k + u. This is going to be k_B+u_B. Now we actually know what the mechanical energy is because we said it always has to equal 8 joules no matter what. So we know this is 8, so all we have to do is just figure out what the potential energy at B in order to figure out what k_B is. So we've got this 8 joules. This equals k_B + 2. So you have 8 minus 2 equals k_B, and therefore you get 6 joules. Going back to our roller coaster analogy, this makes sense. You're going from A, and as you're going from A down to B, you're losing potential energy and therefore you have to gain kinetic energy for your total energy to be 8. So really, the kinetic energy is just going to be the difference between the mechanical energy and the potential at any specific points.

So one way I like to visualize this is by basically looking at the potential energy graph. Here at part A, all of my mechanical potential. Here at part B, I know that I've lost potential and gained some kinetics. So what happens here is that my potential energy at part B is still equal to 2 joules, and the kinetic energy is really just going to be the difference between where I am on the graph and my line of 8 joules. So this vertical line here really just represents my kinetic energy, and I know that this is equal to 6. Let's move on to part c now. Part c we're we're supposed to figure out the speed of the marble at x equals 4. So here I'm back up to x equals 4. So this is going to be my point C, and this is actually very straightforward. We want to figure out the speed at v_C. We're really just going to use our roller coaster analogy. As we're going downhill, we're picking up speed, but then if you're going uphill, you're actually going to lose that speed again. So what ends up happening is, if I started from rest here at point A, and then I'm basically back up to the same heights, if you will, then the speed here at 0 at C also has to be 0. You can't end up going any higher than the initial height from which you started from unless you actually had some initial energy or initial speed, which you didn't in this case. So your speed here at C is going to be 0. And again, this makes sense because basically you can't go anywhere above this 8 joules of energy. Alright? So actually, this brings me to an important conceptual point: you're going to have a speed here at 0, and you're going to have a speed here at 0, And basically, what's going to happen is at this point you're going to go downhill, and this point you're going to go downhill again. So without any additional energy that's added into this problem or removed, the objects are always going to remain stuck underneath this line. They're always going to remain stuck underneath my mechanical energy line, between these two points right here, my v_C or my A and my C. So these are actually called turning points because there are places where the marble is just going to keep turning around forever. Unless it's given some additional energy, it's never going to be able to escape this little sort of well that it's fallen into.

Alright. So now, let's move on to part d. Part d, we're going to figure out without touching the marble again, right, without actually inputting any energy into the system, can it ever reach x equals 5? So x equals 5 is right here, so I'm going to call this point right here point D. So what we know here is that the mechanical energy for this marble throughout the entire problem has always been 8 joules. If we look at the energy that you would require to be at part d, we look across, we look horizontally, and the potential energy you would need is 10 joules. So what ends up happening here is that your mechanical energy, your 8 joules of mechanical energy will always be less than the 10 joules of energy you would need. So the answer to this problem is actually no. You could never actually reach part D. One way I like to think about this also is if you have zero velocity at speed here at part C, and you turned around and went back down the hill again, there's no way you could actually continue upwards and actually arrive at part D. Alright. So that's it for this one, guys. Let me know if you have any questions.

A particle with a mass of 0.1kg moves according to the Potential Energy graph shown. What minimum speed does the particle need at Point A to reach Point B?

• A. 16.3 m/s
• B. 5 m's
• C. 3 m/s
• D. 7.75 m/s

Forces & Equilibrium Positions

Guys, now that we've been introduced to potential energy graphs, there are a couple more conceptual points that you'll need to know about forces and equilibrium positions by using these graphs. So, we're going to work out this example together. Let's check this out. The idea here is that in potential energy graphs, you can gather some information about the sign of the force by looking at the slope of the graph. The sign of f is going to be the opposite sign of the slope. What do I mean by that? Well, let's take a look at our example here. We have a ball that's obeying or following this potential energy graph, and we've got these four points of interest that are labeled here. In part a, we're going to figure out the sign of the force, whether it's just positive, negative, or 0 by looking at these four points here. So, what we're going to do is, I have a, b, c, and d, and to figure this out, I'm just going to look at the slopes at each one of these points here. So, let's take a look at a. Here at a, the tangent line or the slope of the graph is sort of downwards like this. It doesn't have to be perfect; it just, you know, it's really just conceptual. So, the idea here is that the slope of the graph is downwards, and whenever you have downward sloping potential energies, and therefore, the slope is negative, the force is going to have the opposite sign of that. So, the force is going to be positive in this case. So here we have a negative slope, and so we have a positive force. That's the rule. Let's take a look at the second parts. In point b, we're going to have the sort of bottom of this little valley like this. And remember, at the bottoms of the valleys and the tops of the hills, your slope is actually going to be a flat line. So here we have a flat or horizontal slope. And whenever this happens, whenever the slope is flat or horizontal, a horizontal slope means a slope of 0. So therefore, your force is going to be equal to 0. So here we have 0 slope. So therefore, f is equal to 0 here. Alright, that's the answer. So here we've got positive; here we got negative. Let's move on to part c. Part c is basically the opposite of a. So here at point c, your slope is actually going to be upwards like this or positive. So, if your slope is upwards and positive, the force is going to have the opposite sign of that, and it's going to be negative. So here we have a positive slope, therefore, we're going to have a negative force. Now finally, point d is going to be basically the same thing as point b. We have the top of a hill, so therefore, your slope here is going to be flat like this. If you have a flat slope, it's basically you're just going to be a zero force. So we're just going to copy this thing over like this, and that's going to be your force. Alright? So let's take a look now at points b and c. We're going to figure out the positions of stable and unstable equilibrium. What does that mean? Well, remember that when your force is equal to 0, we had a special name for that. That was called equilibrium. So just by looking at the potential energy graph here, we can actually get some information about when the force is equal to 0 for an object and when it's at equilibrium. And depending on what the potential energy graph is doing at these points, these equilibriums actually fall into 2 different categories. So there are 2 different types. The first one is called the stable equilibrium. This happens whenever you have the potential energy graph which is at a minimum. So it's basically going to be right over here. So here, this potential energy graph sort of dips down like this. And so, therefore, it's going to have a minimum value right here. One way I like to think about this is that these minimums happen whenever the potential energy graph is curving up. So an unstable equilibrium is actually the opposite of this. An unstable equilibrium happens whenever the potential energy graph has a maximum like it does in this point over here. So this happens whenever the potential energy is curving down. One way I like to think about this is that if you're a stable person, you're likely pretty happy all the time. So this happens whenever you have sort of a smiley in the potential energy graph. If you're an unstable person, you're generally frowning. You're probably frowning all the time and that's usually what's going to happen here. So, you're going to have a frowny face in the potential energy graph at unstable equilibriums. Alright. So the reason these are called stable and unstable, it has to do with what happens when you have objects that are actually at these equilibrium points. So what I like to do is I kind of like to think about a marble that's sitting in a bowl right here at point b. So imagine you had a little bowl, right? And you put a marble inside of it and eventually, it's going to settle down towards the bottom. So, this we know that at the bottom here, the marble's going to be at equilibrium. What happens if you move it from either one of those from that position? If you move it to the left or to the right, what happens is the marble always wants to return back down to the bottom of the bowl. So the reason it's stable is because if it's ever nudged from this position, objects are always going to return. So they're always going to return back to this position here. An unstable equilibrium is going to be like, if you actually flip the bowl upside-down, right? You flip the bowl upside down, and then you put the marble right on top. If you're able to perfectly balance the marble on top, the marble's going to be at equilibrium here. But what happens if you nudge it from that position? Well, if you nudge it, then the marble just goes flying_off the box or off the bowl like this, and it never can get back up to the top. So what happens is objects will never return back to these equilibrium positions once they're displaced or nudged from those places. So, that's basically what stable versus unstable means. So, to solve parts b and c really quickly here, your positions of stable equilibrium are going to be part point b. Because the f is equal to 0 and we have the curving up. And your unstable equilibrium happens whenever your f is 0 and you're curving down, so that's going to be here point D. So here at point D, your f is equal to 0, that's an equilibrium, but your potential energy graph is curving down like this. So that's really all there is to it for this one, guys. Let's move on.

Potential Energy Graph of a Marble

Alright, folks. So in this problem, we have a marble, and it's moving according to this potential energy function or this potential energy graph. And let's take a look at the first part here, which is we have to figure out the labeled points where the force on the marble is equal to 0. Now remember, the big idea here, guys, is that the force is equal to 0 when the potential energy graph, when U(x), is flat. So that's the most important thing that you need to remember: there's no force when the potential energy graph is flat. So, if you look at this graph here, it's not flat everywhere. There are only 2 points that are flat, 2 of the labeled points, which are points b and d. So that's where the graph momentarily flattens out as it sort of curves, as it changes from curving up to down. So, in other words, the points at which the force is 0 are going to be points b and d. Those are your two points here.

Now let's take a look at the second part here. The second part asks us to calculate which one of these labeled coordinates is a position of stable equilibrium. Remember, we have stable and unstable equilibrium. And remember the rule, use a smiley face. If you're smiling, that means that you're stable. And if you have a sad face, that means that you are unstable. Right? So you want to be stable. Right? That means good. So we're looking for the happy faces. Right? So that means that it's going to be point b. This is also just the one that is stable. This one is unstable which means that if you had a particle, one way you can think about this is if you had a little ball that was at the top of this hill, if it moved a tiny bit to the left or right, it would fall down the hill. But if you had a ball at the bottom of this bowl, it would just stay there. Right? It would always stay there unless something jostled it out of place.

Alright, guys. That's it for this one.

Here’s what students ask on this topic:

How do you calculate the total mechanical energy of an object using a potential energy graph?

The total mechanical energy (E) of an object is the sum of its kinetic energy (K) and potential energy (U). On a potential energy graph, which plots potential energy (U) on the y-axis and position (x) on the x-axis, you can determine the total mechanical energy at a specific point by adding the kinetic energy and the potential energy at that point. If the object is at rest, the kinetic energy is zero, and the total mechanical energy is equal to the potential energy at that position. The equation is given by:

$E=K+U$

What does the slope of a potential energy graph indicate about the force acting on an object?

The slope of a potential energy graph indicates the direction of the force acting on an object. The force (F) is related to the slope of the potential energy (U) graph by the equation:

$F=-\frac{\mathrm{dU}}{\mathrm{dx}}$

A negative slope (downward) means the force is positive, while a positive slope (upward) means the force is negative. A flat slope indicates that the force is zero, corresponding to an equilibrium position.

What is the difference between stable and unstable equilibrium in potential energy graphs?

In potential energy graphs, stable equilibrium occurs at points where the potential energy is at a minimum. If an object is displaced from this point, it will return to the equilibrium position. This is analogous to a marble at the bottom of a bowl. Unstable equilibrium occurs at points where the potential energy is at a maximum. If an object is displaced from this point, it will move away from the equilibrium position. This is like a marble balanced on top of an inverted bowl. Stable equilibrium is characterized by a 'smiley' curve (concave up), while unstable equilibrium is characterized by a 'frowny' curve (concave down).

How can you determine the kinetic energy of an object at a specific position using a potential energy graph?

To determine the kinetic energy (K) of an object at a specific position using a potential energy graph, you need to know the total mechanical energy (E) of the system. The kinetic energy can be found by subtracting the potential energy (U) at that position from the total mechanical energy. The equation is:

$K=E-U$

For example, if the total mechanical energy is 8 joules and the potential energy at the position is 2 joules, the kinetic energy would be:

$K=8-2=6$

Can an object reach a position on a potential energy graph where the required potential energy exceeds its total mechanical energy?

No, an object cannot reach a position on a potential energy graph where the required potential energy exceeds its total mechanical energy. The total mechanical energy (E) is conserved and is the sum of kinetic energy (K) and potential energy (U). If the potential energy required at a certain position is greater than the total mechanical energy, the object does not have enough energy to reach that position. For example, if the total mechanical energy is 8 joules, and the potential energy required at a position is 10 joules, the object cannot reach that position.