If a wave y(x, t)=(6.0 mm) sin(k x+(600 rad / s) t+ϕ) travels along a string, how much time does

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If a wave y(x, t)=(6.0 mm) sin(k x+(600 rad / s) t+ϕ)...

If a wave $y(x, t)=(6.0 \mathrm{mm}) \sin (k x+(600 \mathrm{rad} / \mathrm{s}) t+\phi)$ travels along a string, how much time does any given point on the string tak
to move between displacements $v=+2.0 \mathrm{mm}$ and $y=-2.0 \mathrm{mm} ?$

Key Concepts

Sinusoidal Waves

A sinusoidal wave is a smooth, periodic oscillation represented by a sine or cosine function. Its key parameters include amplitude, which measures the maximum displacement; angular frequency, determining how many oscillations occur per unit time; and phase, which shifts the wave in time or space. Sinusoidal waves are fundamental in describing oscillatory motion such as vibrations on a string.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes oscillatory motion where the restoring force acting on an object is proportional to its displacement from equilibrium and directed toward that equilibrium position. In the context of a sinusoidal wave on a string, each point performs SHM, meaning its displacement follows a sine or cosine function with time.

Angular Frequency and Period

Angular frequency, typically denoted by ?, defines how rapidly the phase of the wave changes with time and is measured in radians per second. It is intrinsically linked to the period of the wave, which is the time taken for one complete cycle of motion, via the relation T = 2?/?. This relationship is crucial for determining time intervals between specific displacement positions.

Phase and Time Relationship

The phase of a wave combines the contributions of time and space in its argument, allowing one to identify specific points on the wave corresponding to particular displacements. Changes in displacement correspond to changes in the phase, and knowing the angular frequency permits converting a phase difference into a time difference. This concept is essential for calculating how long a point takes to move between specified displacement values.

Transcript

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