Physics

Physics

Problem

An experimenter sets up a horizontal mass–spring oscillator on a near-frictionless air track. The cart of mass 0.60 kg0.60\,\text{kg} oscillates with amplitude 0.25 m0.25\,\text{m} about the equilibrium position x=0x=0. At a certain instant the cart is observed to be at x=0.10 mx=0.10\,\text{m} moving toward the equilibrium position. (a) Calculate the numerical value of the fraction of the total mechanical energy that is kinetic at the observed instant. (b) Determine the magnitude of the cart’s speed at that same instant if the period of oscillation is 0.80 s0.80\,\text{s}. (c) Interpret physically why the fraction obtained in part (a) depends only on the ratio x/Ax/A and not on the mass or spring constant.
Intuitive explanation

The key to this problem is energy conservation in simple harmonic motion. At any instant, the total mechanical energy splits between kinetic and potential energy: Etotal=12mv2+12kx2E_{\text{total}} = \frac{1}{2}mv^2 + \frac{1}{2}kx^2. Since you know the amplitude (maximum displacement), you can find the total energy, then use the given position to find how much energy is kinetic—which gives you the velocity you're looking for.

Intuitive explanation

The key to this problem is energy conservation in simple harmonic motion. At any instant, the total mechanical energy splits between kinetic and potential energy: Etotal=12mv2+12kx2E_{\text{total}} = \frac{1}{2}mv^2 + \frac{1}{2}kx^2. Since you know the amplitude (maximum displacement), you can find the total energy, then use the given position to find how much energy is kinetic—which gives you the velocity you're looking for.

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