A Copper plate is soldered between two steel plates. All the plates have the same cross- section (A ) and length l. The coefficient of thermal expansions are and their young’s moduli are respectively. What force will arise in the plates if the temperature is increased by t0 c ? (Assume that the plates suffer the same net expansion)

Thermal expansion of . Elastic contraction of Net expansion of . Either steel plate is subjected to the tensile force from the sides of the cu plate .Net expansion of steel The plus sign is on account of the fact that there is elastic expansion, of steel plates. Since the plates suffer same net expansion

A Copper plate is soldered between two steel plates. All the plates have the same cross- section (A ) and length l. The coefficient of thermal expansions are and their young’s moduli are respectively. What force will arise in the plates if the temperature is increased by t0 c ? (Assume that the plates suffer the same net expansion)

Thermal expansion of . Elastic contraction of Net expansion of . Either steel plate is subjected to the tensile force from the sides of the cu plate .Net expansion of steel The plus sign is on account of the fact that there is elastic expansion, of steel plates. Since the plates suffer same net expansion

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A Copper plate is soldered between two steel plates. All the plates have the same cross- section (A ) and length l. The coefficient of thermal expansions are ${\alpha _c}and\,{\alpha _s}\$ and their young’s moduli are ${y_c}\,and\,{y_s}\$ respectively. What force will arise in the plates if the temperature is increased by t⁰ c ? (Assume that the plates suffer the same net expansion)

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$\frac{{A{y_c}{y_s}\left( {{\alpha _c} - {\alpha _s}} \right)}}{{{y_s} + {y_c}}}\$

$\frac{{A{y_c}{y_s}\left( {{\alpha _c} - {\alpha _s}} \right)}}{{2{y_s} + {y_c}}}\$

$\frac{{2A{y_c}{y_s}\left( {{\alpha _c} - {\alpha _s}} \right)}}{{2{y_s} + {y_c}}}\$

$\frac{{2A{y_c}{y_s}\left( {{\alpha _c} - {\alpha _s}} \right)}}{{{y_s} + {y_c}}}\$

answer is A.

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Detailed Solution

Thermal expansion of $cu = \ell {\alpha _c}t\$ . Elastic contraction of $cu = \frac{F}{A} \times \frac{{\ell _{}^{}}}{{{y_c}}}\$ Net expansion of $cu = \ell {\alpha _c}t - \frac{F}{A}\frac{\ell }{{{y_c}}}\$ . Either steel plate is subjected to the tensile force $\frac{F}{2}\$ from the sides of the cu plate .Net expansion of steel $=\ell {\alpha _s}t + \frac{F}{{2A}} \times \frac{l}{{{y_s}}}\$ The plus sign is on account of the fact that there is elastic expansion, of steel plates. Since the plates suffer same net expansion

$\therefore \ell {\alpha _c}t - \frac{F}{A}.\frac{{\ell _{}^{}}}{{{y_c}}} = \ell {\alpha _s}t + \frac{F}{{2A}}.\frac{\ell }{{{y_s}}}\,\,solving\,\,we\,\,get\,\,F = \frac{{2A{y_c}{y_s}\left( {{\alpha _c} - {\alpha _s}} \right)}}{{2{y_s} + {y_c}}}$