The difference in length between two rods A a | vedclass

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(B) Let the lengths of rods $A$ and $B$ at $0^{\circ}C$ be $\ell_A$ and $\ell_B$ respectively.Given that the difference in length is constant at all t

Vedclass · Accepted Answer

$\ell_{A} = 180 \ cm, \ell_{B} = 120 \ cm$.

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(B) Let the lengths of rods $A$ and $B$ at $0^{\circ}C$ be $\ell_A$ and $\ell_B$ respectively.Given that the difference in length is constant at all temperatures,the change in length for both rods must be equal for any temperature change $\Delta T$.$\Delta \ell_A = \Delta \ell_B$$\ell_A \alpha_A \Delta T = \ell_B \alpha_B \Delta T$$\ell_A \alpha_A = \ell_B \alpha_B$$\ell_A (18 	imes 10^{-6}) = \ell_B (27 	imes 10^{-6})$$\ell_A / \ell_B = 27 / 18 = 3 / 2$So,$\ell_A = 1.5 \ell_B$.Given that the difference in length is $60 \ cm$,we have $\ell_A - \ell_B = 60 \ cm$.Substituting $\ell_A = 1.5 \ell_B$ into the equation:$1.5 \ell_B - \ell_B = 60 \ cm$$0.5 \ell_B = 60 \ cm$$\ell_B = 120 \ cm$.Then,$\ell_A = 1.5 	imes 120 \ cm = 180 \ cm$.Thus,the lengths are $\ell_A = 180 \ cm$ and $\ell_B = 120 \ cm$.

The difference in length between two rods $A$ and $B$ is $60 \ cm$ at all temperatures. If $\alpha_A = 18 \times 10^{-6} /^{\circ}C$ and $\alpha_B = 27 \times 10^{-6} /^{\circ}C$,then the lengths of rod $A$ and rod $B$ at $0^{\circ}C$ are respectively:

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