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Question: A bimetallic strip of brass and steel, each having thickness of 1cm is kept at \({{20}^{\circ }}C\)....

A bimetallic strip of brass and steel, each having thickness of 1cm is kept at 20∘C{{20}^{\circ }}C. The strip bends with the brass on the outer side. The radius of curvature of the curved surface is (αBRASS=18×10−6/∘C, αSTEEL=12×10−6/∘C)({{\alpha }_{BRASS}}=18\times {{10}^{-6}}{{/}^{\circ }}C\text{, }{{\alpha }_{STEEL}}=12\times {{10}^{-6}}{{/}^{\circ }}C)
a) 16.67m
b) 16.67cm
c) 16.67mm
d) 3.33m

Explanation

Solution

In the above question we are basically asked to calculate the radius of curvature when there is a change in temperature. Assume that the change in temperature is 100∘C{{100}^{\circ }}C. Therefore we will use the expression for the radius of curvature of the bimetallic strip to determine the radius.

Formula used:
r=d(a1−a2)ΔTr=\dfrac{d}{({{a}_{1}}-{{a}_{2}})\Delta T}

Complete step-by-step answer:
Let us say a bimetallic strip is made of two metals whose coefficients of thermal expansion are a1 and a2{{a}_{1}}\text{ and }{{a}_{2}} respectively. Let us say further we subject this strip to a change in temperature ΔT\Delta T. This will result in a bent in the strip. Let the thickness of each metal bed. Therefore the radius of curvature for the bent metallic strip is given by,
r=d(a1−a2)ΔTr=\dfrac{d}{({{a}_{1}}-{{a}_{2}})\Delta T}
In the above question the thickness of each of the metals is given to us as 1cm. The coefficient of thermal expansion of brass as well as the steel is given to us, hence the radius of curvature from the above expression we get as,
r=d(aBRASS−aSTEEL)ΔT ⇒r=1×10−2(18×10−6−12×10−6)100 ⇒r=1×10−2(6×10−6)100 ⇒r=1×10−2(6×10−4) ⇒r=1×1026=0.1666×102 ⇒r=16.67m \begin{aligned} & r=\dfrac{d}{({{a}_{BRASS}}-{{a}_{STEEL}})\Delta T} \\\ & \Rightarrow r=\dfrac{1\times {{10}^{-2}}}{(18\times {{10}^{-6}}-12\times {{10}^{-6}})100} \\\ & \Rightarrow r=\dfrac{1\times {{10}^{-2}}}{(6\times {{10}^{-6}})100} \\\ & \Rightarrow r=\dfrac{1\times {{10}^{-2}}}{(6\times {{10}^{-4}})} \\\ & \Rightarrow r=\dfrac{1\times {{10}^{2}}}{6}=0.1666\times {{10}^{2}} \\\ & \Rightarrow r=16.67m \\\ \end{aligned}

So, the correct answer is “Option A”.

Note: It is to be noted that all the physical quantities have to be expressed in the SI units or else the multiplier of the answer obtained will be different. It is given that the bimetallic strip consists of brass on its outer surface. Therefore we have taken α1=αBRASS{{\alpha }_{1}}={{\alpha }_{BRASS}} and α2=αSTEEL{{\alpha }_{2}}={{\alpha }_{STEEL}}.