Question

A surveyor’s 30 m steel tape is correct at a temperature of20°C . The distance between two points, as measured by this tape on a day when the temperature is35°C is 26 m. The true distance between the points is (α_steel=1.2×10−5/°C)

Answer 1

★ Concept :-

Here the concept Linear Expansion has been used. We see that we are given the coefficient of linear expansion. The temperature has been increased from the initial temperature. This means there is a increase in length of the body. Final length is given to us. And according to formula of linear expansion we can get relationship of true length and increased length.

Let's do it !!

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★ Formula Used :-

$\;\boxed{\sf{\pink{\Delta l\;=\;l_{2}\:\alpha\Delta T}}}$

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★ Solution :-

Given,

» Initial Length of Tape = 30 m

» Intial Temperature = T₂ = 20° C

» Final Length of tape = L₁ = 26 m

» Final Temperature = T₁ = 35° C

» Coefficient of expansion of steel = α = 1.2 × 10¯⁵ / °C

» Change in Temperature = ∆T = T₁ - T₂ = 35 - 20 = 15° C

• Let the true length of the tape be L₂

» Change in Length = ∆L = L₁ - L₂ = 26 - L₂

Now let's use Formula to find the answer. For Linear Expansion, we know that

$\;\sf{\rightarrow\;\;\Delta l\;=\;l_{2}\:\alpha\Delta T}$

Now by applying the values, we get

$\;\sf{\rightarrow\;l_{1}\:-\:l_{2}\;=\;l\;=\;l_{2}\:\alpha\Delta T}$

On rearranging, we get

$\;\sf{\rightarrow\;l_{1}\;=\;l_{2}\:\alpha\Delta T\:+\:l_{2}}$

Now taking the common term, we get

$\;\sf{\rightarrow\;l_{1}\;=\;l_{2}(\alpha\Delta T\:+\:1)}$

This will give us,

$\;\sf{\rightarrow\;l_{2}\;=\;\dfrac{l_{1}}{(\alpha\Delta T\:+\:1)}}$

Now putting the numerical values, we get

$\;\sf{\rightarrow\;l_{2}\;=\;\dfrac{26}{(1.2\:\times\:10^{-5}\:\times\:15)\:+\:1}}$

$\;\sf{\rightarrow\;l_{2}\;=\;\dfrac{26}{(18\:\times\:10^{-5})\:+\:1}}$

$\;\sf{\rightarrow\;l_{2}\;=\;\dfrac{26}{(0.00018)\:+\:1}}$

$\;\sf{\rightarrow\;l_{2}\;=\;\dfrac{26}{1.00018}}$

$\;\bf{\rightarrow\;l_{2}\;=\;29.9953\:\;m}$

This is the required answer.

$\;\;\underline{\boxed{\tt{True\;\:length\;\:of\;\:Tape\;=\;\bf{\purple{29.9953\;\;m}}}}}$

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★ More to know :-

• Area Expansion :-

$\;\tt{\leadsto\;\;\Delta A\;=\;A_{2}\:\beta\Delta T}$

• Volumetric Expansion :-

$\;\tt{\leadsto\;\;\Delta V\;=\;V_{2}\:\gamma\Delta T}$

Answer 2

hope is help you dear friend