Physics

Physics

Problem

A 0.20 kg0.20\,\text{kg} cart is attached to a horizontal spring of constant 80 N m−180\,\text{N\,m}^{-1} on a nearly frictionless track. The spring is compressed 0.15 m0.15\,\text{m} from equilibrium and the cart is released from rest. (a) Calculate the total mechanical energy stored in the cart–spring system immediately after release. (b) After several oscillations, non-conservative effects reduce the amplitude to 0.050 m0.050\,\text{m}. Determine the new total mechanical energy at that moment. (c) Compute the percentage decrease in total mechanical energy between parts (a) and (b). (d) Identify one plausible physical mechanism—other than a change in the spring constant—that could account for the energy loss you calculated, and briefly describe how it removes mechanical energy from the system.
Intuitive explanation

This is a classic energy conservation problem where all the spring's stored potential energy converts into the cart's kinetic energy. The key insight is that you don't need to track forces or acceleration—just recognize that the elastic potential energy 12kx2\frac{1}{2}kx^2 at maximum compression equals the kinetic energy 12mv2\frac{1}{2}mv^2 when the spring passes through equilibrium. Set them equal and solve for the velocity.

Intuitive explanation

This is a classic energy conservation problem where all the spring's stored potential energy converts into the cart's kinetic energy. The key insight is that you don't need to track forces or acceleration—just recognize that the elastic potential energy 12kx2\frac{1}{2}kx^2 at maximum compression equals the kinetic energy 12mv2\frac{1}{2}mv^2 when the spring passes through equilibrium. Set them equal and solve for the velocity.

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