Question

A horizontal spring with a spring constant $100N/m$is compressed $20cm$and used to launch a $2.5kg$ box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is $0.15$. Use work and energy to find how far the box slides across the rough surface before stopping.

Short answer

The distance the box slides on the rough surface before stopping is$53.3cm.$

Step-by-step solution

Given information

A horizontal spring with spring constant $100N/m$is compressed$20cm$and is used to launch$2.5kg$box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction between the box and the rough surface is$0.15.$

Explanation

The frictional force acting on a sliding mass mon a surface with a coefficient of kinetic friction $\mu is\mu mg$. Energy is stored in the spring with a spring constant k when it is compressed by a distance $xis\frac{1}{2}k{x}^2$.

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

localid="1648540878531" $$\begin{gathered} E=\frac{1}{2}k{x}^2 \\ E=\frac{1}{2}\times \left(100N/m\right){\left(0.20m\right)}^2 \\ E=2J \end{gathered}$$

This energy is transferred to the mass and used in doing work against the friction force. Work done by friction is

localid="1648541084595" role="math" $$\begin{gathered} W=Force\times dis\tan ce \\ W=\left(\mu mg\right)d=E \\ d=\frac{E}{\mu mg} \\ d=\frac{2J}{\left(0.15\right)\left(2.5Kg\right)\left(9.81m/se{c}^2\right)} \\ d=0.533m \end{gathered}$$

The energy is conserved. This energy stored in the spring is transferred to the mass and used in doing work against the friction force.