Directions: In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:
A bimetallic strip is formed act of two identical strips one of copper and the other of brass. The coefficients of linear expansion of the two metals are\[{{\alpha }_{c}}\]and\[{{\alpha }_{\beta }}\]. On heating, the temperature of the strip goes up by\[\Delta T\]and the strip bends to form an arc of radius of curvature R. Then R is: (1) inversely proportional to\[\Delta T\] (2) proportional to\[|{{\alpha }_{\beta }}-{{\alpha }_{c}}|\] (3) proportional to\[\Delta T\] (4) inversely proportional to\[|{{\alpha }_{\beta }}-{{\alpha }_{C}}|\]A) 1 and 2 are correct
B) 2 and 3 are correct
C) 1 and 4 and correct
D) 1, 2 and 4 are correct
Correct Answer: C
Solution :
Let to be the initial length of each strip before heating. Length after heating will be \[{{l}_{B}}={{l}_{0}}(1+{{\alpha }_{\beta }}\Delta T)=(R+d)\theta \] and \[{{R}_{c}}={{l}_{0}}(1+{{\alpha }_{c}}\Delta T)=R\theta \] \[\therefore \] \[\frac{R+d}{R}=\frac{1+{{\alpha }_{\beta }}\Delta T}{1+{{\alpha }_{c}}\Delta T}\] \[\therefore \] \[1+\frac{d}{R}=1+({{\alpha }_{\beta }}-{{\alpha }_{c}})\Delta T\] [from binomial expansion] \[\therefore \] \[R=\frac{d}{({{\alpha }_{\beta }}-{{\alpha }_{c}})\Delta T}\] Or \[R\propto \frac{1}{\Delta T}\]and \[\propto \frac{1}{|{{\alpha }_{\beta }}-{{\alpha }_{c}}|}\]You need to login to perform this action.
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