Question

A$1000kg$safe is $2.0m$above a heavy-duty spring when the rope holding the safe breaks. The safe hits the spring and compresses it $50cm$. What is the spring constant of the spring?

Short answer

The spring constant of the spring is$k=156,960N/m$.

Step-by-step solution

Given information

The weight of the safe$=1000kg$

The distance between the safe and heavy-duty spring, $h=2m$

The safe hits the spring and compresses it $50cm$.

Explanation

The potential energy of a mass at height h is taken as $mgh$. Energy stored in the spring with spring constant k when it is compressed by a distance x is $\frac{1}{2}k{x}^2$.

The safe hits the spring with KE,

localid="1649411903428" $$\begin{gathered} KE=mgh \\ KE=\left(1000kg\right)\times \left(9.81m/s^2\right)\times \left(2m\right) \\ KE=19620J \end{gathered}$$

This energy is stored in the spring when it is compressed. That is

localid="1649411942679" $$\begin{gathered} KE=\frac{1}{2}k{x}^2 \\ 19,620J=\frac{1}{2}k{\left(0.50m\right)}^2 \\ k=\frac{2\times \left(19,620J\right)}{{\left(0.50m\right)}^2} \\ k=156,960N/m \end{gathered}$$

The energy is conserved. The potential energy of the safe is converted to the energy stored in the spring when the spring is compressed.