 Let's solve a couple of questions on applying work energy principle when there is a graph of force versus position. So for the first one, we have a horizontal net force F which is exerted on an object at rest. The object starts at x equals to 0 meters and has a speed of 6 meters per second after moving 3 meters along a horizontal frictionless surface. The net force F as a function of the object's position is shown below. Here we have the graph force versus position. We need to figure out the mass of the object. As always, why don't you pause the video and first try this one on your own. Alright, hopefully you have given us a shot. Now here we need to figure out the mass of the object and the information that we have in the question. Let's try to list down the pieces of information that we know. We know that there is an object at rest to begin with. So let's let's draw a ball and this is it is at rest initial speed of 0. The object starts at x equals to 0 and has a speed of 6 meters per second after moving 3 meters along a horizontal frictionless surface. So when the object is over here and let's say this distance, this is 3 meters. After 3 meters, the speed, the speed is 6 meters per second. The speed is 6 meters per second. And there is a force as well which is exerted on the object at rest. Force is constantly increasing and at this point there is a greater force when the object is at a 3 meter, 3 meter distance. Let's try to see that from the graph when the object is at a 3 meter distance. This is 3, the force, force would be 15 newtons. So this would be 15 newtons when the object is at a 3 meter position. And the speed at this point is 6 meters per second. Okay, so what is happening in this situation? Object is starting from rest. Now it has some kinetic energy, right? Because it has some speed. So initial kinetic energy is 0, now it has some kinetic energy. There is also a force and there is also displacement. That means there is some work being done, right? So all of this work that is being done on the ball, this is leading, this is resulting into a gain in kinetic energy. So this is resulting into a gain in kinetic energy or a change in kinetic energy. In this case, the speed is increasing. So we can say gain because it's starting from 0 to some value. And this right here is really, this is the work energy principle. So all the work that is being done on this ball because of the force that is acting on the ball is resulting into a gain of kinetic energy. Now when you think about the work being done, let's try to figure out what is the work being done. We have a graph here, force versus position. For the first 3 meters, it's not a constant force. The force is constantly increasing, right? There also be acceleration involved. If there is acceleration, there must be some velocity also. But the force over here, it's not a constant. So how do we figure out the work being done for the first 3 meters? Well, for a force position graph, we can calculate this area, this area of a triangle. That is, this is area of the triangle. And this is equal to the change in kinetic energy. So the final kinetic energy, that is, that is half into m into 6 square minus half into m into 0, initial is 0. And the area of triangle, this is half into base into height. So that is half into base into height. And this is half into 3 into 50. This is equal to half into m into 36. 36 is 6 square divided by 2, 18, 18 m. So if we work this out, m comes out to be equal to, well, let's say 15 into 3 is 45. And this 18 comes down as well. So this is 45 divided by 36. This can be, this is multiples of, multiples of 9. So this is 5 and 4. And this will come out to be equal to 1.25 kilograms. Now we need to round the answer to two significant digits. So we can express this as 1.3 kilograms. And that is a mass. That is a mass of the object. Okay, let's look at one more question. Here we have a horizontal net force, which is exerted on a 1 kilogram object at rest. We have very similar things. The object starts at 0 and it travels 5 meters along a horizontal frictionless surface. And we have the force versus position graph, which is very similar to the previous question. We need to figure out the velocity this time, not the mass. Please do attempt this on your own before you see how I approach it. Okay, hopefully we have given this a shot. Now the situation seems to be very similar, right? So we can again start with the ball and initial is initial starting from rest at rest. This is 1 kilogram. We know the mass and it travels 5 meters along a horizontal frictionless surface. So this position right here, we can say that this, this is 5 meters. It has some velocity now. We do not know what that is. Mass of course is 1 kilogram. And we again know the force, the force that is, that is acting on this ball. When the object is at a position of 5 meters, let's see the force. That is, that will be, that will be 25, 25, 25 Newtons. So if there is a force and displacement, there will be work. And here, the force and the displacement, they are in the same direction. So the object is also gaining some velocity, right? Because there is a force, which means it is gaining some kinetic energy, there is some change in kinetic energy. Where is it coming from? From the work being done. So again, the work energy principle, the work being done, this is equal to change in kinetic energy. Now let's apply this. Again, let's try to figure out the area, area of this triangle. This is a work being done, that is half into base, 5 into height, 25. And that is equal to the change in kinetic energy. Initial is 0, final is half into mass, which is 1, into u square. We do not know what u is. So when we work this out, this comes out to be equal to 5 into 25. That is 125, yeah, 125. This is equal to u square. This is equal to u square. So u is under root of 125. Now what is this? We need to calculate this. When we work this out, this is 11.18 meters per second. And we need to round the answer to 3 significant digits. So this would be 11.2. Alright, you can try more questions from this exercise in the lesson. And if you are watching on YouTube, do check out the exercise link, which is added in the description.