Question

A metal rod that is $30.0cm$ long expands by $0.0650cm$ when its temperature is raised from $0^{\circ }C$ to ${100}^{\circ }C$ . A rod of a different metal and of the same length expands by $0.0350cm$ for the same rise in temperature . A third rod , also $30.0cm$ long , is made up of pieces of each of the above metals placed end-to-end and expands $0.0580cm$ between $0^{\circ }C$ and ${100}^{\circ }C$ . Find the length of each portion of the composite bar .

Answer 1

$l_1=30cm$

$d{l}_1=0.065cm$

$dT={100}^{\circ }C$

$dl=\alpha \ast L\ast dT$

${\alpha }_1=\frac{d{l}_1}{L_{1}dT}$

by putting the values we get, ${\alpha }_1=2.167\times {10}^{-5}$

$l_2=30cm$

$d{l}_1=0.065cm$

${\alpha }_2=\frac{d{l}_2}{L_{2}dT}$

by putting the values we get, ${\alpha }_2=1.167\times {10}^{-5}$

Let one position be $x$ with ${\alpha }_1$

The other position length be, $30-x$ with ${\alpha }_2$

$dl=2.167\times {10}^{-5}\times x\times 100+1.167\times {10}^{=5}(30-x)\times 100$

By solving the above equation we get $x=20cm$

The other part is $(30-20)=10cm$