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Problem 42 Hard Difficulty

An object with mass $m$ moves with position function
$$\mathbf{r}(t)=a \sin t \mathbf{i}+b \cos t \mathbf{j}+c t \mathbf{k} \quad 0 \leqslant t \leqslant \pi / 2$$
Find the work done on the object during this time period.

Answer

Work done $=\frac{m\left(b^{2}-a^{2}\right)}{2}$

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Video Transcript

So in this video, given the position Vector rt's well to be scientific, I have plus B coastline TJ plus CTK happened. We're told that t goes from zero to pi. We're also there's another way where we can write our of tea and a contact way, the fallen way Any scientific of a Because I see to And, uh, now, uh, derivative of r is just a co sign t carbon minus B certainty. I see. And then finally, if we take our double price, well, we will notice. Is that that's just a negative. A sign t native be co sign t at the curative over Constant to know Uh, of course, in this questions were asked to determine forced We know that the force acting on a particle from your second was just f time f is equal to me where m is the mass and pays the exploration. But remember, the exploration is the second derivative of the position. So we're given the position back. There was the second derivative. We determined it to be negative. A sign t call a negative B coast trying to become zero. So we just got a multiply this right here. Bite him. We're just multiplying by a constant. So here, what we're gonna do, They distribute this and and then what we get is that the force is negative. M a sign T. I have ah minus and be co sign T J Hot zero K. Huh? So this is divorced again. We can write it in a different way. It is. Okay. Since we're asked to determine the work, we know that the work is just the integral off the curveball about your and then now we're going to just write everything in terms of t. So this is the integral from zero it apart over two of our prime. All right, but this is just a formula the famous into All right, So we were determined. F so f is just negative. Any sign t called the negative and Vico side becomes zero. We determined our prime, which just acres 90 color negative. Be scientific on zero, and then t know we're going to take the dot product. So we're gonna multiply first term first term. Here it's second term. Second of the third. All right, So what we get is integral from zero toe by over two negative Emmy squared scientist, Go 70 plus and be swear it signed people 70. So our third term that because we're multiplying by zero. All right, now what we can pull up. So now we're going to do something that's a bit tricky. So we see here we have I saw a sign Ko Sai and signed Close I we're gonna pull that out of it as a common factor. And what we're left with is m b squared, minus Emini square. And just another thing that might save weird right now, we're also gonna pull a common factor, too. So since there is a two here, we're gonna put it to here, All right? No, no. Why didn't we do this? Well, this whole this term right here is a constant which we could pull to the outside. And since we have scientific society, we know our famous trick identity that to sign Tico scientist assigned to t. So now we get this dice integral right here. Where m b squared minus eight square divided by two is just a constant. We pulled to the outside and when we have signed to T, what we can do is you can use our your substitution, So we're gonna let u equal to two t so two year becomes to DT. So DT's just deal divided by two. All right, so now this is our new integral and we changed our limits of integrations are limits of integration work for equals zero to tease equals pi over two. But now we change that. So we're gonna multiply zero byte to so he gives me room. We're going to multiply pi over two by two. So we get time, all right? And this too Just a constant which we can pull to the outside. So now we get A and B squared minus east were divided by four because people that have to be outside And then now what's the integral of sine of you, Bobby? Integral of sine of use. Just negative co sign of you on our limits of integration are now from zero up I so again Ah, we saw this Integral really. Plug in pie for its negative co sign of pi minus negative co sign of zeros are becomes a postscript Sign of zero. Well, co signer flies just negative one. There's a negative on the outsides of this becomes a positive one. Co sign of zero is warns that one plus one is two. So this entire things too. So we're multiplying by two dividing by four. We can simplify this So we're left with m is s So we're left with The work is equal to them times being squared when this space where invited to Okay?

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