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A steel tape 1m long is correctly calibrated for a temperature of ${{27}^{0}}C$. The length of a steel rod measured by this tape is found to be 63.0cm on a hot day when the temperature is ${{45}^{0}}C$. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is${{27}^{0}}C$ ? Coefficient of linear expansion of steel $1.20\times {{10}^{-5}}{{K}^{-1}}$

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Last updated date: 23rd May 2024
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Answer
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Hint: Let us first find the new length of the steel tape when the temperature is changed to given temperature. Once the new length is known, we can calculate the length of steel rod at a new temperature by multiplying the steel rod length with the ratio of new and old lengths of steel tape.

Formula used: $\Delta l=l\alpha \Delta T$

Complete step by step answer:
length of the steel tape at temperature $T={{27}^{0}}C,l=100cm$
At temperature ${{T}_{1}}={{45}^{0}}C$, the length of the steel rod, ${{l}_{1}}=63cm$
Coefficient of linear expansion of steel, $1.20\times {{10}^{-5}}{{K}^{-1}}$
Let l be the actual length of the steel rod and l' be the length of the steel tape at 45oc.
$\begin{align}
  & \Delta l=l\alpha \Delta T \\
 & \Rightarrow {{l}^{1}}-l=l\alpha (45-27) \\
 & \Rightarrow {{l}^{1}}=100+100\times 1.2\times {{10}^{-5}}\times 18 \\
 & \Rightarrow {{l}^{1}}=100.0216cm \\
\end{align}$
Hence, the actual length of the steel rod measured by the steel tape at ${{45}^{0}}C$ can be calculated as:
$\begin{align}
  & {{l}_{2}}=\dfrac{100.0216}{100}\times 63 \\
 & \Rightarrow {{l}_{2}}=63.0136cm \\
\end{align}$
therefore, the actual length of the rod at ${{45}^{0}}C$ is $63.0136cm$.
Its length at ${{27}^{0}}C$ is $63cm$.

Additional Information: Thermal expansion is the tendency of matter to change its shape, which could be the length, area, density or volume when there’s a change in the temperature of the surroundings. This process usually doesn’t include any phase transitions. Temperature is a monotonic function of the average molecular kinetic energy of a substance. There are three types of thermal expansions, one is linear expansion, second one is a real expansion or superficial expansion, the third one is called the cubical expansion or the volumetric expansion.

Note: In the above question, the length of the tape is changed when there's an increase in the temperature but not an change in the area or total volume. So, linear thermal expansion formula is used. Also, the measured length will be changed as the tape and rod are made of the same material and both expand at the given temperature accordingly.
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