Question

An iron rod is 1.58 m long at 0°C. What must
be the length of a brass rod at 0°C if the
difference between the lengths of the two rods
is to remain the same at all temperatures.
Linear expansivity of iron = 1.2 x 10-5 K-1
Linear expansivity of brass = 1.9 x 10-5 K-1.​

please this urgent

Answer 1

The length of the brass rod must be 1.00 m

Explanation:

The change in length of a metal rod due to the increase in temperature is given by

$\Delta L = L_0 \alpha \Delta T$

where

$L_0$ is the initial length of the rod

$\alpha$ is the linear expansivity of the metal

$\Delta T$ is the increase in temperature

For the iron rod, we can write the following

$\Delta L = L_i \alpha_i \Delta T$

Where

$L_i = 1.58 m$ is the initial length of the iron rod

$\alpha_i = 1.2\cdot 10^{-5} K^{-1}$ is the iron linear expansivity

For the brass rod, we can write

$\Delta L = L_b \alpha_b \Delta T$

Where

$L_b =$ is the initial length of the brass rod

$\alpha_b = 1.9\cdot 10^{-5} K^{-1}$ is the brass linear expansivity

We want the two changes in length to be the same for the same change in temperature $\Delta T$, so we can write

$L_i \alpha_i \Delta T=L_b \alpha_b \Delta T$

And solving for $L_b$, we find the length of the brass rod:

$L_b = \frac{L_i \alpha_i}{\alpha_b}=\frac{(1.58)(1.2\cdot 10^{-5})}{1.9\cdot 10^{-5}}=1.00 m$

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