A rail tack made of steel having length $$l0$$ m is clamped on a railway line at its two ends (figure). On a summer day due to rise in temperature by $$20$$°C. It is deformed as shown in figure. Find x (displacement$$ of the $$centre) if α$$steel=$$ $$1.2$$ ×$$10-5/$$ $$0C$$

Question

A rail tack made of steel having length $$l0$$ m is clamped on a railway line at its two ends (figure). On a summer day due to rise in temperature by $$20$$°C. It is deformed as shown in figure. Find x (displacement$$ of the $$centre) if                       α$$steel=$$ $$1.2$$ ×$$10-5/$$ $$0C$$

Infinity Learn · Accepted Answer

Consider the diagram.Applying Pythagoras theorem in right angled triangle in $$figure(L+$$∆$$L2)2$$ $$=$$ (L2)2$$ $$+x2$$⇒x $$=$$ $$(L+$$∆$$L2)2-(L2)2$$   $$=$$ $$12(L+$$∆$$L)2-L2$$⇒$$12($$∆$$L2+2L$$∆$$L)As$$ increase in length ∆L is very small, therefore, neglecting $$(($$∆$$L)2), we getx $$=$$ $$12$$×$$2L$$∆L    $$---------(i)But$$ ∆L $$=$$ L α∆$$t------(ii)Substituting$$ value of ∆L in Eq.(i) from Eq.$$(ii)x=$$ $$122L$$×Lα∆t $$=$$ $$12L2$$α∆t    $$=$$ $$102$$×$$2$$×$$1.2$$×$$10-5$$×$$20$$    $$=$$ $$5$$×$$4$$×$$1.2$$×$$10-4$$ $$=$$ $$2030$$ ×$$10-3$$ cm

A rail tack made of steel having length l0 m is clamped on a railway line at its two ends (figure). On a summer day due to rise in temperature by 20°C. It is deformed as shown in figure. Find x (displacement of the centre) if αsteel= 1.2 ×10-5/ 0C

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