A 300 dollars g oscillator has a speed of 95.4 cm / s when...

A 300 dollars g oscillator has a speed of $95.4 \mathrm{cm} / \mathrm{s}$ when its displacement is $3.0 \mathrm{cm}$ and $71.4 \mathrm{cm} / \mathrm{s}$ when its displacement is $6.0 \mathrm{cm} .$ What is the oscillator's maximum speed?

Key Concepts

Velocity-Displacement Relationship in SHM

The relationship between velocity and displacement in simple harmonic motion is given by v = ??(A² - x²), where v is the speed at displacement x, ? is the angular frequency, and A is the amplitude. This equation illustrates how the speed varies with position: it is maximum at the equilibrium point (x = 0) and zero at the turning points (x = A).

Determining Maximum Speed from Experimental Data

Using measured speeds at specific displacements enables the determination of the system parameters such as amplitude and angular frequency by applying the energy conservation equation or the velocity-displacement relationship. Once the amplitude and angular frequency are known, the maximum speed, which is given by ?A, can be computed. This methodology is key in analyzing oscillatory systems and understanding their dynamic properties.

Simple Harmonic Motion (SHM)

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position. This motion is characterized by oscillatory behavior with a sine or cosine function, and it serves as a fundamental model in mechanics for systems like springs and pendulums.

Energy Conservation in SHM

In simple harmonic motion, the total mechanical energy, composed of kinetic and potential energy, remains constant. At any given displacement, the energy is partitioned between kinetic energy (which is highest at equilibrium where displacement is zero) and potential energy (which is highest at the maximum displacement, or amplitude). This principle allows one to relate the speed of the oscillator to its displacement.

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