question_answer

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 \[\propto \frac{1}{|{{\alpha }_{\beta }}-{{\alpha }_{c}}|}\]