The equation of a transverse wave travelling on a rope is...

The equation of a transverse wave travelling on a rope is given by $y=10 \sin \pi(0.01 x-2.00 t)$ where $y$ and $x$ are in $\mathrm{cm}$ and $t$ in seconds. The maximum transverse speed of a particle in the rope is about [MP PET 1999] (a) $63 \mathrm{~cm} / \mathrm{sec}$ (b) $75 \mathrm{~cm} / \mathrm{s}$ (c) $100 \mathrm{~cm} / \mathrm{sec}$ (d) $121 \mathrm{~cm} / \mathrm{sec}$

The equation of a transverse wave travelling on a rope is given by $y=10 \sin \pi(0.01 x-2.00 t)$ where $y$ and $x$ are in $\mathrm{cm}$ and $t$ in seconds. The maximum transverse speed of a particle in the rope is about [MP PET 1999]
(a) $63 \mathrm{~cm} / \mathrm{sec}$
(b) $75 \mathrm{~cm} / \mathrm{s}$
(c) $100 \mathrm{~cm} / \mathrm{sec}$
(d) $121 \mathrm{~cm} / \mathrm{sec}$

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Key Concepts

Amplitude and Angular Frequency

Amplitude is the maximum displacement from the equilibrium position, defining the energy of the wave, while angular frequency measures how rapidly the wave oscillates in time. Together, they are key parameters that characterize the wave's behavior.

Differentiation in Wave Analysis

Differentiation of the wave function with respect to time is used to determine the particle velocity in the medium. The maximum transverse speed of a particle is obtained by taking the time derivative of the displacement function, highlighting the role of calculus in analyzing dynamic systems.

Transverse Waves

Transverse waves are disturbances where the displacement of the medium is perpendicular to the direction in which the wave travels. This concept is fundamental in understanding the behavior of waves in various media such as ropes, strings, and electromagnetic fields.

Wave Equation

The wave equation is a mathematical expression that describes the spatial and temporal evolution of wave phenomena. It typically involves sinusoidal functions to model the displacement of the medium as a function of both position and time.

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