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

"> 2}}={{(PR)}^{2}}-{{(PO)}^{2}}={{l}^{2}}-{{\left( \frac{l}{2} \RIGHT)}^{2}}\] \[{{[l(1+{{\ALPHA }_{2}}t)]}^{2}}-{{\left[ \frac{1}{2}(1+{{\alpha }_{1}}t) \right]}^{2}}\] \[{{l}^{2}}-\frac{{{l}^{2}}}{4}={{l}^{2}}(1+{{a}_{2}}^{2}{{t}^{2}}+2{{\alpha }_{2}}t)-\frac{{{l}^{2}}}{4}(1+{{a}_{1}}^{2}{{t}^{2}}+2{{\alpha }_{1}}t)\] 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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Three rods of equal length / are joined to form an equilateral triangle PQR. 0 is the mid point of R 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

NEET Physics in NEET 1 year ago

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[d] \[{{(OR)}^{2}}={{(PR)}^{2}}-{{(PO)}^{2}}={{l}^{2}}-{{\left( \frac{l}{2} \RIGHT)}^{2}}\] \[{{[l(1+{{\ALPHA }_{2}}t)]}^{2}}-{{\left[ \frac{1}{2}(1+{{\alpha }_{1}}t) \right]}^{2}}\] \[{{l}^{2}}-\frac{{{l}^{2}}}{4}={{l}^{2}}(1+{{a}_{2}}^{2}{{t}^{2}}+2{{\alpha }_{2}}t)-\frac{{{l}^{2}}}{4}(1+{{a}_{1}}^{2}{{t}^{2}}+2{{\alpha }_{1}}t)\] 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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