A rail track made of steel having length 10 m is clamped on a railway line at its two ends as shown in 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 alpha_("steel")=1.2xx10^(-5)" "^(@)C^(-1)

A rail track made of steel having length 10 m is clamped on a railway

1.	A rail track made of steel having length 10 m is clamped on a railway line at its two ends as shown in 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 alpha_("steel")=1.2xx10^(-5)" "^(@)C^(-1)
Answer» SOLUTION :By using Pythagoras theorem, for given right angle triangle, `((L+DELTAL)/(2))^(2)=(L/2)^(2)+x^(2)` `x=SQRT(((L+DeltaL)/(2))^(2)-(L/2)^(2))` `=1/2(sqrt((L+DeltaL)^(2)-L^(2)))` `=1/2[sqrt(L^(2)+2LDeltaL+DeltaL^(2)-L^(2))]` `=1/2[sqrt(2LDeltaL+DeltaL^(2))]` As `DeltaL` is very small, `DeltaL^(2)` is neglected `x=1/2sqrt(2LDeltaL)` But `DeltaL=LalphaDeltaT` `:.x=1/2sqrt(2LxxLalphaDeltaT)` `=1/2Lsqrt(2alphaDeltaT)` `:.x=10/2xxsqrt(2xx1.2xx10^(-5)xx20)` `=5xxsqrt(4.8xx10^(-4))` `=5xx2xx1.1xx10^(-2)` `=11xx10^(-2)` `=0.11m=11cm` Here, the INCREASE in length of mercury is neglected as it is very small.