Question

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

Short answer

The length of each portion A and B of rod is 32.0 cm and 7 cm respectively.

Step-by-step solution

Concept/Significance of

step 1 is missing in the document.

Determination of formula for the length of rods

The change in length of the metal rods is given by,

$∆L=∆L_A+∆L_B$

Here, $∆L_A$is the change in length of rod A and $∆L_B$is the change in the length of B.

The value of $∆L_A$is given by,

$∆L_A={\alpha }_A{L}_A∆T$

Here, ${\alpha }_A$is the coefficient of expansion, $L_A$is the length of rod A and $∆T$is the change in temperature.

Similarly,

$∆L_B={\alpha }_B{L}_B∆T$

Here, ${\alpha }_B$is the coefficient of expansion, $L_B$is the length of rod B and $∆T$is the change in temperature.

Determination of the value of length of each portion of the composite rod

the length equation the value of $L_A$and $L_B$is given by,

$L_A=\frac{∆L/∆T-\left(-0.300\right){\alpha }_B}{{\alpha }_A-{\alpha }_B}$

And

$L_B=30cm-L_A$

Substitute all the values in the above two equations.

$$\begin{gathered} L_A=\frac{\left(0.058\times {10}^{-2}m\right)/\left(100°C\right)-\left(0.300m\right)\left(1.65\times {10}^{-5}{\left(°C\right)}^{-1}\right)}{1.00\times {10}^{-5}{\left(°C\right)}^{-1}} \\ =23.0cm \\ {L}_B=30cm-23cm \\ =7cm \end{gathered}$$

Thus, the length of each portion A and B of rod is 23.0 cm and 7 cm respectively.