A simple harmonic oscillator has spring constant k=5.0 N / m amplitude A=10 cm, and maximum sp

step 1 We're given an oscillator that has a spring constant of 5 newtons per meter, an amplitude of osc

A simple harmonic oscillator has spring constant k=5.0 N /...

A simple harmonic oscillator has spring constant $k=5.0 \mathrm{~N} / \mathrm{m}$ amplitude $A=10 \mathrm{~cm},$ and maximum speed $4.2 \mathrm{~m} / \mathrm{s} .$ What's the oscillator's speed when it's at $x=5.0 \mathrm{~cm} ?$

Key Concepts

Angular Frequency

Angular frequency is a crucial parameter in simple harmonic motion that quantifies how rapidly the system oscillates. It relates to the system’s physical parameters and links the maximum speed with the amplitude via the relation v_max = ?A. This concept also facilitates the connection between displacement and velocity throughout the oscillation.

Simple Harmonic Motion

Simple harmonic motion describes the oscillatory motion of a system in which the restoring force is directly proportional to the negative of the displacement. This type of motion results in periodic, sinusoidal oscillations and is characterized by parameters such as amplitude, period, and frequency, which succinctly capture the system’s dynamics.

Conservation of Energy in Oscillatory Systems

In systems undergoing simple harmonic motion, mechanical energy is conserved when non-conservative forces (like friction) are negligible. The total energy is the sum of kinetic and potential energies. At maximum displacement, the energy is entirely potential, whereas at the equilibrium position, it is entirely kinetic. This energy balance enables the determination of the velocity at any given displacement.

Velocity-Displacement Relationship in SHM

The velocity at any point in simple harmonic motion can be derived from energy conservation, leading to a relationship of the form v² = ?²(A² - x²). This equation expresses how the kinetic energy (and therefore the speed) depends on both the amplitude of the motion and the instantaneous displacement, allowing one to find the speed at any position along the oscillatory path.

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