Question

Write down the expression for the elastic potential energy of a stretched wire.

Answer 1

Hint: We know that the potential energy held by stretching or compressing an elastic object by an external force, such as stretching a spring, is known as elastic potential energy. It is equal to the work required to extend the spring, which is proportional to the spring constant \[k\] and the length of the spring.

Formula used:
\[dW = F \cdot dl\]
Here, $dW$ is the work done, $F$ is the force and $dl$ is the small displacement.
\[Y = \dfrac{F}{A} \times \dfrac{L}{I}\]
Here, $\dfrac{F}{A}$ is the force per unit area and $\dfrac{L}{I}$ fractional change in length.

Complete step by step solution:
Now we know that the elastic potential energy is equal to the work required to extend the spring, which is proportional to the spring constant \[k\] and the length of the spring. So, the work done in stretching the wire \[dl\] is,
\[dW = F \cdot dl\]

Now the total work done in stretching the wire from \[0\] to \[l\] is
\[W = \int_0^l {F \cdot dl...\left( 1 \right)} \]
Now we know from the young’s modulus elasticity that our force becomes
\[Y = \dfrac{F}{A} \times \dfrac{L}{I}\]
\[\Rightarrow F = \dfrac{{YAI}}{L}...\left( 2 \right)\]
By substituting the equation \[\left( 2 \right)\] in \[\left( 1 \right)\], we get
\[W = \int_0^l {\dfrac{{YAI}}{L}} dl\]

Since \[I\] is the dummy variable in the integration, we can change \[I\] to \[I'\] as shown below:
Therefore, \[W = \int_0^l {\dfrac{{YAI'}}{L}} dl\]
\[W = \dfrac{{YA}}{L}\left[ {\dfrac{{I{'^2}}}{2}} \right]_0^l\]
\[\Rightarrow W = \dfrac{{YA}}{L} \cdot \dfrac{{{I^2}}}{2}\]
\[\Rightarrow W = \dfrac{1}{2}\left[ {\dfrac{{YAI}}{L}} \right]I\]
Further solving we get,
\[W = \dfrac{1}{2}FI\]
Thus, this work done is also known as the elastic potential energy of a stretched wire.

Hence, the expression for the elastic potential energy of a stretched wire is \[W = \dfrac{1}{2}FI\].

Note: Students should be aware that within elastic bounds, young's modulus can also be defined as the ratio of longitudinal stress to longitudinal strain. Now, the elastic limit is the greatest stress at which a body can restore its original shape and size after a deforming force is removed. For issue solving, all of these minor definitions are required.