A light rod of length $$3m$$ is suspended from the ceiling horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of Aluminium and is of $$cross-section$$ $$6$$×$$10$$−$$3m2$$ and other is of copper of cross section $$2$$×$$10$$−$$3m2$$ . Find the position along the rod at which a weight may be hung to produce $$i)$$ Equal stresses in both wires and YAℓ$$=7$$×$$1010N/m2$$ $$ii)$$ Equal strain in both wires $$Ycu=11$$×$$1010N/m2$$

Question

A light rod of  length $$3m$$ is suspended from the ceiling horizontally by means of two vertical  wires of equal length tied to its ends. One of the wires is made of Aluminium  and is of $$cross-section$$ $$6$$×$$10$$−$$3m2$$     and other is of  copper of cross section $$2$$×$$10$$−$$3m2$$     . Find the position along the rod at which a weight may be  hung to produce  $$i)$$ Equal stresses in both wires and YAℓ$$=7$$×$$1010N/m2$$     $$ii)$$ Equal strain in both wires $$Ycu=11$$×$$1010N/m2$$

Infinity Learn · Accepted Answer

$$i)$$ Equal $$stressa1=6$$×$$10$$−$$3m2$$ Here $$a2=2$$×$$10$$−$$3m2$$⇒$$T1a1=T2a2$$ ⇒$$T1T2=a1a2=6$$×$$10-32$$×$$10-3=3$$ ∴$$T1=3T2$$................(1)In$$ equilibrium, moment of all the forces acting on the rod about a point is zero.Let the centre of gravity acts at a distance of x from string $$1$$ and $$3-x$$ from string $$2$$                 $$T1x=T2(3$$−$$x)----(2) from (1) and (2)  $$3T2x=T2(3$$−$$x)$$ $$3x=3-x$$ $$4x=3$$ $$x=34=0.75m$$                     $$ii)$$ Equal strain, $$e2=e2T1a1y1=T2a2y2$$, $$T1T2=a1y1a2y2=6$$×$$10-3$$×$$7$$×$$10102$$×$$10-3$$×$$11$$×$$1010$$⇒$$T1T2=2111$$....................(3)From$$ $$(2) and (3) we get $$T1x=T2(3-x)$$ ⇒$$T1T2=(3-x)x$$ ⇒$$2111=(3-x)x$$ ⇒$$21x=33-11x$$ ⇒$$32x=33$$  ⇒$$x=3332=1.03$$ m

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