A system in which only one particle moves has the potential energy shown in FIGURE EX10.31. What is the x-component of the force on the particle at x = 5, 15, and 25 cm?

Video transcript

Hey everyone. So this problem is working with potential energy graphs. Let's see what it's asking us. A single particle is moving in an environment that is characterized by a potential energy graph. As shown below, determine the X component of the force acting on the particle. When it is positioned at three different positions, X equals three centimeters, six centimeters and 10 centimeters. So this is a three part question. Let's go through our potential answer choices. We've got a part one negative newtons, 20 newtons, 3 500 newtons B -250 Newtons, zero Newtons, 50 Newtons, C 150 newtons, 10 newtons, 500 newtons or D 250 newtons, zero newtons and 100 newtons. OK. So the first thing that we can recall here is that force in terms of potential energy is given by the equation F equals negative D U over D X. And the derivative of a function we can recall is the slope of the function where slope is rise overrun. Uh Another way to write that is the change in Y over the change in X. So I'm just gonna say like the derivative or the slope is delta Y over delta X. And so we can rewrite this force equation as negative delta U over delta X, which can be simplified to negative UF -UI four, final potential energy minus initial potential energy over final position minus initial position or X F minus X I. So we're gonna, we are going to use this equation for all three parts of the problem. So first up, we have X equals three cm on this graph. We can see That at x equals three cm. We are along this first line here. Now, the slope of this line is constant, But we don't know the exact value when X is at three cm, we do however know the value or when X is at 4cm Here, the value is gonna be 10 jewels. And because the slope is constant, we can use that point rather than the point at three cm. And it's going to be the same slope. So that looks like F X equals negative 10 Jews minus zero Joles because we're going from the point from 0 to centimeters is from zero Joles to 10 jules and then four cm. I'm going to rewrite as 0.04 m. It's not negative. So 0.04 m minus zero m. And we plug that in to our calculator And we get negative 250 newtons. So that is the x component of the force acting on the particle when it is at 3cm. So we could look at our potential answers and we can eliminate answer choice C&D or X equals six centimeters. That's part two, we can see here that we have a horizontal line Where x equals six cm. So there's no change in potential energy. Therefore, that first term delta U is zero. So the force is zero. That is our answer. For part two, we can see actually that both of our, the choices left have the correct answer for part two. So last thing we need to do here is figure out part three where X equals 10 centimeters. So again, 10 cm here, The uh slope of this point from eight cm, the slope of this line from eight cm to 10 cm is constant. And so we can use What we know at the values that we know at eight cm and the values we know at 10 cm to solve for our force. So f of X or 10 cm our final potential energy. So at 10 cm is zero, Jews, our initial potential energy. So for this portion of the line, that's at eight centimeters, that's 10 Jews. So we have zero Jews, I'm sorry, negative zero Jews minus 10 Jews. That's all divided by our final position, which is 10 centimeters. I'm gonna rewrite that as 100.1 m just to keep everything in uh standard units Minour our initial point, which was eight cm. Again, that's 0.08 m. We plug that into our calculator And we get newtons And that is the correct answer for part C or Part three. So then when we go back to our potential answer choices that aligns with answer choice A. So that's all we have for this one. We'll see you in the next video.