Spiderman uses his spider webs to save a runaway train moving about \(60\;{{km} \mathord{\left/{\vphantom {{km} h}} \right.} h}\)Fig. 6–44. His web stretches a few city blocks (500 m) before the \({\bf{1}}{{\bf{0}}^{\bf{4}}}\;{\bf{kg}}\) train comes to a stop. Assuming the web acts like a spring, estimate the effective spring constant.

FIGURE 6–44Problem 72.

The value of the spring constant can be identified by analyzing the value of the spring force and the spring's deflection.

Its value alters inversely to the value of the spring deflection for a constant value of force.

Given data:

The initial speed of the train is \({v_1} = {\rm{60}}\;{{{\rm{km}}} \mathord{\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}\).

The spring deflection is\(x = 500\;{\rm{m}}\).

The mass of the train is\(m = {10^4}\;{\rm{kg}}\).

It is given that the train is stopped at the final position. So, its final velocity will be \({v_2} = 0\).

The work that Spiderman needs to do to stop the train is equal to the change in the kinetic energy of the train. Therefore, the value of the work done can be calculated as:

\(\begin{aligned}W &= \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2\\W &= \frac{1}{2}\left( {{{10}^4}\;{\rm{kg}}} \right){\left( 0 \right)^2} - \frac{1}{2}\left( {{{10}^4}\;{\rm{kg}}} \right){\left( {\left( {60\;{{km} \mathord{\left/{\vphantom {{km} h}} \right.} h}} \right)\left( {\frac{{1000\;{\rm{m}}}}{{1\;{\rm{km}}}}} \right)\left( {\frac{{{\rm{1}}\;{\rm{h}}}}{{{\rm{3600}}\;{\rm{s}}}}} \right)} \right)^2}\\W &= - 1.39 \times {10^6}\;{\rm{J}}\end{aligned}\)

Here, the negative sign shows that Spiderman does the work against the train.

The work done by Spiderman is equal to the potential energy of the spring. So, the spring constant can be calculated as:

\(\begin{aligned}W &= - \frac{1}{2}k{x^2}\\k &= - \frac{{2W}}{{{x^2}}}\\k &= - \frac{{2\left( { - 1.39 \times {{10}^6}\;{\rm{J}}} \right)}}{{{{\left( {500\;{\rm{m}}} \right)}^2}}}\\k &= 11.1\;{{\rm{N}} \mathord{\left/{\vphantom {{\rm{N}} {\rm{m}}}} \right.} {\rm{m}}}\end{aligned}\)

Thus, the effective spring constant is \(11.1\;{N \mathord{\left/{\vphantom {N m}} \right.} m}\).