Two rods of equal length of 60 cm each are joined together end to end. Coefficient of linear expansions of the rods are 24 × 10^-6 °C^-1 and 1.2 × 10^-5 °C^-1. Their temperatures are same and equal to 30°C which is increased to 100°C. Find final length of the combination (in cm).

The correct option is 120.1512.

Step-by-step Explanation:

When the temperature of a rod is increased, its length changes due to linear thermal expansion. The relation between the final length and initial length is given by the formula:

$L_{\text{final}}=L_0(1+\alpha \Delta T)$

where:
•
$L_0$
is the initial length of the rod,
•
$\alpha$
is the coefficient of linear expansion, and
•
$\Delta T$
is the change in temperature.

For a combination of two rods joined end to end, the total final length of the combination is the sum of the final lengths of the two individual rods:

$L_{\text{total}}=L_{\text{1,final}}+L_{\text{2,final}}$

Let us write down the given parameters:
• Initial length of both rods:
$L_1=L_2=60\text{cm}$
• Coefficient of linear expansion for the first rod:
${\alpha }_1=24\times {10}^{-6}{\text{°C}}^{-1}$
• Coefficient of linear expansion for the second rod:
${\alpha }_2=1.2\times {10}^{-5}{\text{°C}}^{-1}=12\times {10}^{-6}{\text{°C}}^{-1}$
• Initial temperature:
$T_i=30\text{°C}$
• Final temperature:
$T_f=100\text{°C}$

First, we calculate the change in temperature (
$\Delta T$
):

$\Delta T=T_f-T_i=100\text{°C}-30\text{°C}=70\text{°C}$

Now, let us calculate the expansion (change in length,
$\Delta L$
) of each rod.

For the first rod:

$\Delta L_1=L_1{\alpha }_1\Delta T$

$\Delta L_1=60\times (24\times {10}^{-6})\times 70$

$\Delta L_1=100800\times {10}^{-6}=0.1008\text{cm}$

For the second rod:

$\Delta L_2=L_2{\alpha }_2\Delta T$

$\Delta L_2=60\times (12\times {10}^{-6})\times 70$

$\Delta L_2=50400\times {10}^{-6}=0.0504\text{cm}$

Now, let us calculate the total change in length (
$\Delta L_{\text{total}}$
):

$\Delta L_{\text{total}}=\Delta L_1+\Delta L_2=0.1008\text{cm}+0.0504\text{cm}=0.1512\text{cm}$

Finally, the total length of the combination is:

$L_{\text{total}}=(L_1+L_2)+\Delta L_{\text{total}}$

$L_{\text{total}}=(60+60)+0.1512=120.1512\text{cm}$