Compute the work done in moving a particle from the point 𝑃(1,−1,0)𝑡𝑜𝑄(2,0,1)by application of force field 𝐹=2𝑥𝑖+3𝑦𝑗+4𝑧𝑘&nbsp;

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Compute the work done in moving a particle from the point 𝑃(1,−1,0)𝑡𝑜𝑄(2,0,1)by application of force field 𝐹=2𝑥𝑖+3𝑦𝑗+4𝑧𝑘&amp;nbsp;

RizerMix · Accepted Answer

We are given a force field F→=2xi→+3yj→+4zk→ and we need to compute the work done in moving a particle from point P(1,−1,0) to Q(2,0,1).To calculate the work done, we use the formula:W=∫CF→·dr→where r→ is the position vector of the particle along the path C from P to Q.First, we need to find the parametric equation of the line segment that connects P to Q. We can do this by using the vector equation of a line:r→(t)=P→+t(Q→−P→)where P→=&amp;lt;1,−1,0&amp;gt; and Q→=&amp;lt;2,0,1&amp;gt;.Substituting the values, we get:r→(t)=&amp;lt;1,−1,0&amp;gt;+t(&amp;lt;1,1,1&amp;gt;)Simplifying, we get:r→(t)=&amp;lt;1+t,−1+t,t&amp;gt;Next, we need to calculate dr→, which is the differential of r→ with respect to t.dr→=&amp;lt;dx,dy,dz&amp;gt;dr→dt=&amp;lt;1,1,1&amp;gt;Therefore, dr→=&amp;lt;dx,dy,dz&amp;gt;=&amp;lt;dt,dt,dt&amp;gt;=dt&amp;lt;1,1,1&amp;gt;Now, we can substitute these values into the formula for work:W=∫CF→·dr→=∫tPtQF→(r→(t))·dr→where tP and tQ are the values of t that correspond to points P and Q on the path, respectively.Substituting the values, we get:W=∫01F→(r→(t))·dr→=∫01(2(1+t)i→+3(−1+t)j→+4tk→)·(dt&amp;lt;1,1,1&amp;gt;)=∫01(2+2t−3+3t+4t)dt=∫01(9t−1)dt=92t2−trvert01=92(1)−1−92(0)+0=72Therefore, the work done in moving the particle from P to Q by the force field F→ is 72.

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Compute the work done in moving a particle

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