If the equation of a wave in a string having linear mass...

If the equation of a wave in a string having linear mass density $0.04 \mathrm{~kg} \mathrm{~m}^{-1}$ is given by $\mathrm{y}=0.02$ $\sin [2 \pi\{1 /(0.04)\}-\{\mathrm{x} /(0.50)\}]$, then the tension in the string is $\ldots \ldots \ldots \ldots$ N. (All values are in $\mathrm{mks}$ ) (A) $6.25$ (B) $4.0$ (C) $12.5$ (D) $0.5$

If the equation of a wave in a string having linear mass density $0.04 \mathrm{~kg} \mathrm{~m}^{-1}$ is given by $\mathrm{y}=0.02$ $\sin [2 \pi\{1 /(0.04)\}-\{\mathrm{x} /(0.50)\}]$, then the tension in the string
is $\ldots \ldots \ldots \ldots$ N. (All values are in $\mathrm{mks}$ )
(A) $6.25$
(B) $4.0$
(C) $12.5$
(D) $0.5$

Key Concepts

Tension in a String

Tension (T) in a string is a force that acts along its length and is necessary to support wave propagation. The tension affects the wave speed through the relationship T = ?v², where v is the wave speed and ? is the linear mass density. This equation is used to determine the tension in the string when the wave speed and mass density are known.

Wave Equation Parameters

This concept involves understanding the structure of a sinusoidal wave, which is typically expressed as y = A sin(?t ? kx) or a similar form. The parameters include amplitude (A), angular frequency (?), period (T = 2?/?), wavelength (?), and wave number (k = 2?/?). Recognizing these parameters from the wave equation allows one to extract the time and spatial characteristics of the wave propagation.

Wave Speed

Wave speed is a fundamental property defined as the distance a wave travels per unit time. It can be calculated either by the relationship v = ?/T, where ? is the wavelength and T is the period, or by v = ?/k, where ? is the angular frequency and k is the wave number. This measure is crucial because it relates the temporal and spatial aspects of wave motion.

Linear Mass Density

Linear mass density (?) is the mass per unit length of a string or cable, which is a critical parameter when analyzing wave motion along a stretched string. It affects the dynamics of the wave; a higher mass density results in slower wave propagation for a given tension, while a lower density allows for faster propagation.

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