Single Variable Calculus, Chapter 6, 6.4, Section 6.4, Problem 14

Use the Riemann Sum to show how to approximate the required work. Then express the work as integral and evaluate it. Suppose that a chain lying on the ground has a length from $10 m$ long and mass $80 kg$. How much work is required to raise one end of the chain to a height of $6m$?

Recall from Hooke's Law:

$f(x) = kx$ where $f$ is force, $k$ is spring constant and $x$ is the maximum elongated length.

From Newton's Law,


$
\begin{equation}
\begin{aligned}

& ma = kx
\\
\\
& 80 kg \left( 9.8 \frac{m}{s^2} \right) = k (10m)
\\
\\
& k = 78.4 \frac{N}{m}

\end{aligned}
\end{equation}
$



Notice that the part of the chain $x$ refers from the lifted end is raised $(6 - x)$ refers for $0 \leq x \leq 6$ meters, and it is lifted meters for $x > 6$ meters. So the work done using the Riemann Sum is..

$\displaystyle W = \lim_{n \to \infty} \sum_{i = 1}^n 78.4 (6 - xi) \Delta x$

If we evaluate the integral, we have..


$
\begin{equation}
\begin{aligned}

& W = \int^6_0 78.4 (6-x) dx
\\
\\
& W = 78.4 \left[ 6x - \frac{x^2}{2} \right]^6_0
\\
\\
& W = 1411.2 \text{ Joules}

\end{aligned}
\end{equation}
$