Question

A brass wire of length $1.8m$ long at ${{27}^{0}}C$ is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of $-{{39}^{0}}C$, what is the tension developed in the wire, if its diameter is $2.00mm$. Coefficient of linear expansion of brass and Young’s modulus of brass are $2\times {{10}^{-5}}{{K}^{-1}},0.91\times {{10}^{11}}Pa$.

Answer 1

Hint: There are two ways the wire could be changed by length. one is due to the temperature change caused by external means; another way is by applying force at both the ends. The tension exerted or developed on it can be calculated by equalising both the formulas.

Formulas used:
$\begin{align}
  & Y=\dfrac{F}{A}\times \dfrac{l}{\Delta l} \\
 & \Delta l=\alpha l({{T}_{2}}-{{T}_{1}}) \\
\end{align}$


Complete step by step answer:
Let us assume the tension applied or developed as $T$. Given,
initial temperature, length of wire, final temperature, diameter, coefficient of linear expansion and young’s modulus as ${{27}^{0}}C,1.8m,{{39}^{0}}C,2\times {{10}^{-3}}m,2.0\times {{10}^{-5}}{{K}^{-1}},0.91\times {{10}^{11}}Pa$ respectively.
now, young’s modulus can be calculated as,
$\begin{align}
  & Y=\dfrac{F}{A}\times \dfrac{l}{\Delta l} \\
 & \Delta l=\dfrac{Fl}{AY} \\
\end{align}$
the change in the length due to change in temperature is,
$\Delta l=\alpha l({{T}_{2}}-{{T}_{1}})$
Now, as both the change in length are for the same rod, they must be equal,
so,
$\begin{align}
  & \dfrac{Fl}{AY}=\alpha l({{T}_{2}}-{{T}_{1}}) \\
 & F=\dfrac{AY\alpha l({{T}_{2}}-{{T}_{1}})}{l} \\
 & F=(2\times {{10}^{5}})\times (-39-27)\times 3.14\times 0.91\times {{10}^{11}}\times {{10}^{-6}} \\
 & F=-3.8\times {{10}^{2}}N \\
\end{align}$
The reason why a negative sign is placed is because the tension is directed inwards to the wire.

Additional information:
Coefficient of linear expansion is the ratio of increase in length of the wire by one degree rise in temperature. The ratio of increase in area of the wire due to rise of one degree is known as superficial expansion. This concept is used to know how much tension an object can withstand with respect to its shape and size under the influence of heat radiation. This coefficient of expansion is an intrinsic property. Hence, it varies from one material to another material.

Note:
The final tension exerted or developed on the wire or the material is taken negative as the force exerting on the material is inward. If there is no external force that resists the expansion of the material, the tension would be positive only. Also, the temperature is decreased which means the object expands on cooling. Thus, the two rigid supports are not allowing it to expand.