Question

A steel rope has length l, area of cross-section a, young's modulus y. [density=d]. If the steel rope is vertical and moving with the force acting vertically up at the upper end, find the strain at a point $\dfrac{L}{3}$​ from lower end.

Answer 1

Hint: Let us find the tension of the steel rope at a point. As the rope is given a definite and considerable mass, tension changes from one point to another point as the length changes. After that, find the tension at a given distance and that tension is the force acting on the body.

Formula used: $Y=\dfrac{stress}{strain}$

Complete step by step answer:
Let us first find the tension acting on the steel rope. As the steel rope has mass and it is not negligible, the tension changes as the point chosen on the steel rope changes.
If we calculate the tension at the upper most point where steel rope is taught, we can easily find it at the other places too. The tension increases as the length of the point from the upper most point increases.
The tension at upper most point and at given point will be is,
$\begin{align}
  & T\alpha l \\
 & \Rightarrow {{T}_{\dfrac{l}{3}}}=\dfrac{T}{3} \\
\end{align}$
Now, lets us calculate the force acting in terms of young’s modulus and other parameters,
$F=\dfrac{dA\lg }{2}$
As the gravitational force acts at the centre of the rope, the length here is taken as half the total length.
Now, the tension is equal to the force acting, therefore, tension is,

$T=\dfrac{dA\lg }{6}$
Now, we need to find out the strain,
We know,
$\begin{align}
  & \Rightarrow Y=\dfrac{stress}{strain} \\
 & \Rightarrow strain=\dfrac{stress}{Y} \\
 & \Rightarrow strain=\dfrac{F}{AY} \\
 & \Rightarrow strain=\dfrac{dALg}{6AY} \\
 & \Rightarrow strain=\dfrac{d\lg }{6Y} \\
\end{align}$
Therefore, we have derived the strain at required length.

Additional Information: The young’s modulus or also called the modulus of elasticity in tension is a property of mechanical nature that measures the tensile stiffness of a solid material. It quantifies the relationship between tensile stress and the axial strain. This was named after the British scientist Thomas young. The term modulus is derived from the Latin root term meaning measure.

Note: The tension is changing at each and every point because the mass of rope is given and is not negligible. If the mass is negligible, the net tension at the overall rope will be the same. Also, the young’s modulus is the ratio of stress and strain.