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A variable force $F(x)$ in the positive $x$ -direction is graphed in the accompanying figure. Find the work done by the force on a particle that moves from $x=0$ to...

Question

A variable force $F(x)$ in the positive $x$ -direction is graphed in the accompanying figure. Find the work done by the force on a particle that moves from $x=0$ to $x=5$(FIGURE CAN'T COPY)

A variable force $F(x)$ in the positive $x$ -direction is graphed in the accompanying figure. Find the work done by the force on a particle that moves from $x=0$ to $x=5$ (FIGURE CAN'T COPY)



Answers

(II) The force on a particle, acting along the $x$ axis, varies as shown in Fig. 6-38. Determine the work done by this force to move the particle along the $x$ axis: ($a$) from $x =$ 0.0 to
$x =$ 10.0 m; ($b$) from $x =$ 0.0 to $x =$ 15.0 m.

I was looking part and you want to find the work done over the distance is X equals zero. Texaco's 10 meters and work is the area underneath the force versus distance curve. We are given that a the curve in this in this scenario is a trap resort and the formula for the area of a trap. A soda is just the top side, plus the bottom side over to that quantity should be multiplied by the height of the trap aside. So the top side of the trap, the top side of the trap is or has length four meters. Bottom side has 10 meters. You divide that whole thing by two multiplied by the height, which is just 400 Newtons, which is the force that is being applied over the distance. And then, if we were to do the math, we find that the work done from X equals zero to X equals 10 meters is 2800 jewels for part B. We're gonna use a similar approach because it's the curve is also a trap design, except we have a little wrinkle because thief forces also is negative in this case. But this approach is the same. So the top side in what is what's really the bottom side. But we're gonna consider the topside is the has length. Two meters. The bottom side has length five meters. Divide that by two and then the height of this trap is oId is negative 200 newtons, which gives us a work done of negative 700. Jules. But remember, we're not done because this just gives us the work done from X equals 10 2 x equals 15 years, but not over the whole thing. So what we need to do is we need to take the work done from 0 to 10 meters which is 2800 jewels. Subtract or rather add negative 700 jewels to it, which is the same thing is subtracting 700 jewels from it. And then we get 2100 jewels is due 2100 joules of work is done from X equals zero. The meters two X equals 15. You just circle my answers just so that we're clear. All right, so a is 2800 jewels and B is 20

So here we can approximate the work done. If we have a force versus displacement diagram Ah, we can find the area under the curve. And here the area under a curve represents a Travis oId. So we can say that the work done is simply going to be equal to 400 Nunes times 1/2 and then this will be 10 meters plus four meters and this is giving us 2800 jewels. Now for part B, it is again representing another trap. A story, however, will be going from 10 meters, too 15 meters. And so work done is going to be equal to hear b negative 200 Nunes times again 1/2 ln here it would be five meters plus two meters and this is giving us negative 700 jewels and then for part, see, they just want the network done, which would be 2800 minus 700 giving us 2100 jewels. So this would be your answer for it. See, for part B and for part, a sober party and party, you simply have to approximate it as a trap is oId and then first see, you simply have to find the difference between these two. That is the end of the solution. Thank you for watching

Here we fuss. Plot the four sources Displacement forces on y axis and displacement on X axis. Here we have. If Lord, this is zero. And here we have minus F. Lord, this is explored and this is two times off. Ex Lord, if you plot then this goes The line goes from picks mod and meet If Lord here given a one dimensional force therefore fix effects which is a function of X The world is simply equal to the area under the curve. So what will be called to integral off Ex Initial two x file force dx for about a The port is shown here we take explored to be a positive and the world done. He's inactive. There's the object moves from ex physical 202 x Is it going to explode? And the world is a positive as it moves from X physical Do it smart two x is equal to two x amount So this is negative here and here. It's a positive since the 80 off a triangle we know any offer trying will be No, he's the one or two times a based I'm altitude. Then the work done from X more acceptable to 02 X is equal to X. Marred is doubly. One is equal to minus X Lord times if no divided by two and the world done from X p Simple to fix Nod two X is equal to to explode is given by W to that easy Cool too well, um two ex lord minus X terms if f more divided by two. This gives us explored F Lord divided by two. The total work is then some off to So not who is one plus w two on this gives us minus one over to f more explored plus one or two explored f mort to be zero but be the integral for the work Done. Who can be written is a w integration from 0 to 2 Ex Lord, This is f into the function we have If more x divided by explored minus one d x are integrating this we get if more times x squared, divided by two ex Lord minus X integrating from zero to to explode this user stance her to be zero

But a off the given problem The plot moved off. The function function will be following. You have here X axis, which is X displacement. And here's the force, which is function off X. This is a maximum 10 euros. Zero. And here is to, uh yeah, 0.2 divisions Airport, four divisions. And here is the one. Um, this is a 24 mmm 68 since a 10. So the plot off the function will be following the cream's is here and up to do two year estimating the area under the curve Allow lust for the range off the answers. Ah, from from 11 Joel to our 14. Jules, if you're looking, uh, this plot, the work will be called to simply the 80 under this curve, which is a force times, displacement, X, but be we can estimate the area analytically by taking integration from 0 to 2 max exes were given function that is staying. He excellent, too. DX We integrate this, we get the minus 20 e minus X divided too. The limits for the X 0 to 2. So this gives us ah, 12.6 jewels, which is approximately 13. Jules


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