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JEE Main & Advanced Physics Thermometry, Calorimetry & Thermal Expansion Question Bank Critical Thinking

  • question_answer
    Three rods of equal length l are joined to form an equilateral triangle PQR. O is the mid point of PQ. Distance OR remains same for small change in temperature. Coefficient of linear expansion for PR and RQ is same i.e. \[{{\alpha }_{2}}\] but that for PQ is \[{{\alpha }_{1}}\]. Then

    A)             \[{{\alpha }_{2}}=3{{\alpha }_{1}}\]

    B)             \[{{\alpha }_{2}}=4{{\alpha }_{1}}\]

    C)             \[{{\alpha }_{1}}=3{{\alpha }_{2}}\]

    D)            \[{{\alpha }_{1}}=4{{\alpha }_{2}}\]

    Correct Answer: D

    Solution :

                       \[{{(OR)}^{2}}={{(PR)}^{2}}-{{(PO)}^{2}}={{l}^{2}}-{{\left( \frac{l}{2} \right)}^{2}}\] \[={{[l(1+{{\alpha }_{2}}t)]}^{2}}-{{\left[ \frac{l}{2}(1+{{\alpha }_{1}}t) \right]}^{2}}\] \[{{l}^{2}}-\frac{{{l}^{2}}}{4}={{l}^{2}}(1+\alpha _{2}^{2}{{t}^{2}}+2{{\alpha }_{2}}t)-\frac{{{l}^{2}}}{4}(1+\alpha _{1}^{2}{{t}^{2}}+2{{\alpha }_{1}}t)\] Neglecting \[\alpha _{2}^{2}{{t}^{2}}\] and \[\alpha _{1}^{2}{{t}^{2}}\] \[0={{l}^{2}}(2{{\alpha }_{2}}t)-\frac{{{l}^{2}}}{4}(2{{\alpha }_{1}}t)\Rightarrow \,2{{\alpha }_{2}}=\frac{2{{\alpha }_{1}}}{4}\Rightarrow ;\,{{\alpha }_{1}}=4{{\alpha }_{2}}\]


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