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When a particle is located a distance $ x $ meters from the origin, a force of $ \cos (\frac{\pi x}{3}) $ newtons acts on it. How much work is done in moving the particle from $ x = 1 $ to $ x = 2 $? Interpret your answer by considering the work done from $ x = 1 $ to $ x = 1.5 $ and from $ x = 1.5 $ to $ x = 2 $.

0 Joules; See solution for interpretation of answer

Applications of Integration

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This question. We need to find that little walk down given that particles located at a distance six m from urgent. And the forces given by a cause bikes over three newtons which checks on it. Uh we need to find a world from 1 to 2 by interpreting the meaning of the world from 1 to 1.5 and then one from 1.5 to 2. So we know that the work done in case the force is not constant is given by F dot T X. And we need to integrate this within the limits. So it's from 1 to 1.5, it's from 1 to 1.5. So the forces given by a cost by X over three our time's D. X. And this should be integrated within 1 to 1.5. Integration of course a sign and three over pie comes outside. So we have three hour prior times sine of pi X over three is that require integration. The limits are from 1 to 1.5. This comes out of 3/5. This comes out as three or prior times Sine of pi over two minus sign of by over three. So sign 90 is one and science 60 degree is through three or two. So this is the required bogdan, definitely this is positive. So this is the first, let's call it a blue D one. So the world in one And likewise we are going to find the walked into. So the only difference is the limits. So uh we're gonna take it forward from here because the integration is going to remain the same. The only difference is the limits. So that's signed by X over three. Within the limits of 1.5- two. So if you place the a parliament we have sign of two pi over three minus sign of by or were too. It's a sign of two pi over three is nothing but sine of pi minus by over three. Uh There's nothing but rules three or 2 and this is -1. So definitely this is negative. Why is this negative? This is negative. That's because the direction of the force has now changed, the direction of the force has now changed after 1.5 the stump becomes negative. So this is negative because the force has changed it's direction. So that is the reason why uh why the by the water is now negative. So the network done is going to be we are both these things. So the network done uh W. T. Is equal to w network worked on one plus walked into. And clearly the both of these are equal and opposite to each other. So when we add them we are going to get zero. So that's where the network done in zero. Because the first world known as positive and the net and the second world that is negative because they are the direction of the force as Changed in the 2nd 1. Thank you