Answer
427.8k+ views
Hint: To solve this question, we will use the concept of the young’s modulus and the dependence of length or size of material on temperature. Obtain the expression for Young’s modulus and the change in length of the wire with temperature. Put the given values and make required arrangements to find the answer.
Complete answer:
Given I the question that,
The coefficient of linear thermal expansion of the steel wire is $\alpha ={{10}^{-5}}{{/}^{0}}C$
The young’s modulus of the steel wire is, $Y={{10}^{11}}N/{{m}^{2}}$
The original length of the wire is L at ${{T}_{i}}={{40}^{0}}C$
The wire expands in length after we hang a mass m from its free end.
At ${{T}_{f}}={{30}^{0}}C$, the wire regains its original length L with the mass m.
The radius of the wire is $r=1mm={{10}^{-3}}m$
The young modulus of any wire can be defined as the ratio of the stress and the strain produced in the wire.
$\begin{align}
& Y=\dfrac{\text{stress}}{\text{strain}} \\
& Y=\dfrac{\dfrac{mg}{A}}{\dfrac{\Delta L}{L}} \\
& Y=\dfrac{mgL}{A\Delta L} \\
\end{align}$
Now, the change in length can be expressed as,
$\begin{align}
& \Delta L\propto L\Delta T \\
& \Delta L=\alpha L\Delta T \\
\end{align}$
Putting this value on the expression of young’s modulus, we get,
$\begin{align}
& Y=\dfrac{mgL}{A\alpha L\Delta T}=\dfrac{mg}{A\alpha \Delta T} \\
& m=\dfrac{YA\alpha \Delta T}{g} \\
\end{align}$
The value of acceleration due to gravity, $g=9.8m{{s}^{-2}}$
The cross-sectional area of the wire is, $\begin{align}
& A=\pi {{r}^{2}}=\pi \times {{\left( {{10}^{-3}} \right)}^{2}} \\
& A={{10}^{-6}}\pi {{m}^{2}} \\
\end{align}$
The change in temperature of the wire is, $\begin{align}
& \Delta T={{T}_{i}}-{{T}_{f}} \\
& \Delta T={{40}^{0}}C-{{30}^{0}}C \\
& \Delta T={{10}^{0}}C \\
\end{align}$
Putting the given values on the expression for mass, we get,
$\begin{align}
& m=\dfrac{{{10}^{11}}\times {{10}^{-6}}\pi \times {{10}^{-5}}\times 10}{10} \\
& m=\pi \\
& m\approx 3kg \\
\end{align}$
So, the mass m will be nearly 3kg.
The correct option is (A).
Note:
Young’s modulus or the modulus of elasticity of a material can be defined as a mechanical property which measures the stiffness of the material. It gives us the ability of the material to regain its original shape when the object is subjected under a force to deform its shape.
Complete answer:
Given I the question that,
The coefficient of linear thermal expansion of the steel wire is $\alpha ={{10}^{-5}}{{/}^{0}}C$
The young’s modulus of the steel wire is, $Y={{10}^{11}}N/{{m}^{2}}$
The original length of the wire is L at ${{T}_{i}}={{40}^{0}}C$
The wire expands in length after we hang a mass m from its free end.
At ${{T}_{f}}={{30}^{0}}C$, the wire regains its original length L with the mass m.
The radius of the wire is $r=1mm={{10}^{-3}}m$
The young modulus of any wire can be defined as the ratio of the stress and the strain produced in the wire.
$\begin{align}
& Y=\dfrac{\text{stress}}{\text{strain}} \\
& Y=\dfrac{\dfrac{mg}{A}}{\dfrac{\Delta L}{L}} \\
& Y=\dfrac{mgL}{A\Delta L} \\
\end{align}$
Now, the change in length can be expressed as,
$\begin{align}
& \Delta L\propto L\Delta T \\
& \Delta L=\alpha L\Delta T \\
\end{align}$
Putting this value on the expression of young’s modulus, we get,
$\begin{align}
& Y=\dfrac{mgL}{A\alpha L\Delta T}=\dfrac{mg}{A\alpha \Delta T} \\
& m=\dfrac{YA\alpha \Delta T}{g} \\
\end{align}$
The value of acceleration due to gravity, $g=9.8m{{s}^{-2}}$
The cross-sectional area of the wire is, $\begin{align}
& A=\pi {{r}^{2}}=\pi \times {{\left( {{10}^{-3}} \right)}^{2}} \\
& A={{10}^{-6}}\pi {{m}^{2}} \\
\end{align}$
The change in temperature of the wire is, $\begin{align}
& \Delta T={{T}_{i}}-{{T}_{f}} \\
& \Delta T={{40}^{0}}C-{{30}^{0}}C \\
& \Delta T={{10}^{0}}C \\
\end{align}$
Putting the given values on the expression for mass, we get,
$\begin{align}
& m=\dfrac{{{10}^{11}}\times {{10}^{-6}}\pi \times {{10}^{-5}}\times 10}{10} \\
& m=\pi \\
& m\approx 3kg \\
\end{align}$
So, the mass m will be nearly 3kg.
The correct option is (A).
Note:
Young’s modulus or the modulus of elasticity of a material can be defined as a mechanical property which measures the stiffness of the material. It gives us the ability of the material to regain its original shape when the object is subjected under a force to deform its shape.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)