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Sppose that a pointmass 'm'is movingunder a constantforce vecF = 2hati-hatj + hatk netweon .At someinstant , t=0, pointP(xm, ym, -1m) [m- metre ] is the instantaneous position of themass. We knowthattorque can beexpressed as thecross- product of positionvector and forcesvector, i.e., tau= vecr xx vecF . AtP, torquecan beexpessedastau= (-4hatj - 4 hatk)NmAt some other instant,t=3 sec, the pointmasshas anotherinstantaneouspositionQ(x_(1), y_(1), z_(1))suchthatthe displacementvectorsbetweenpoints P and Qand thegivenforce are mutually perpendicular. Also, x-component of torqure at Q is zero and yz-components are equal in magnitude and direction alongthe negativedirectionof therespectiveaxes. Usinga definitescale, if we constructa parallelogram with the positionvectorsof Q and thegivesforce vecF as itsadjacent sides , areaof thisparallelogramis 2sqrt(2)m^(2) . Areaof the given parallelogram , in fact , representsa physicalquantitywhose magnitude in SI systemcan beexpressed as 5times thegives are Answerthe following questions.Coordinates of P are : |
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Answer» `""= ("XI" + "yj"-hatk)xx (2i-j+hatk)` `""vectau_(p)=(y -1)i -(x + 2)j-(x +2y)hatk` Given`""vectau_(p) = - 4hatj - 4hatk` `therefore""x=2 and y=1` ALSO, `""vecr_(PQ)=(x_(1) -x) i+ (y_(1) - y)j +(z_(1) + 1)hatk` Given `""vecr_(PQ) botvecF` `therefore""vecr_(PQ) .vecF = 0` `therefore[(x_(1) - 2)hati + (y_(1) - 1) hatj + ( z_(1)+ 1)hatk].[2i - hatj+hatk]= 0......(i)` `""tau_(Q)= vecr_(QO) xxvecF= |{:("i","j","k"),(x_(1),y_(1),z_(1)),(2,-1,1):}|` `tau_(Q)= (y_(1) + z_(1))hati - j (x_(1) -2z_(1))+ k(-x_(1)- 2y_(1))` Given, `""y_(1) + z_(1) =0""...........(ii)` And,`""x_(1) -2z_(1) =x_(1) + 2y_(1)""......(iii)` Also,`""|vecr_(QO) xx vecF= 10sqrt(2)` `therefore (y_(1) + z_(1))^(2) + (x_(1) -2z_(1))^(2) + (x_(1) + 2y_(1))^(2) = 100 xx 2....(iv)` `therefore ""2(x_(1) - 2z_(1))^(2) = 100 xx 2` `therefore""x_(1) - 2z_(1) = 10` `therefore""vectau_(Q)= - 10 hatj - 10 hatk` `""W = 0, " as" vecr_(PQ)` is prependicular to `vecF` |
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