The equation y(x,t)=A\cos 2\pi f(xv−t) may be written as y(x,

erurnSopSoypegx

erurnSopSoypegx

Answered question

2021-12-06

The equation y(x,t)=Acos2πf(xvt) may be written as y(x,t)=Acos[2πλ(xvt)].
Use the last expression for y(x,t) to find an expression for the transverse velocity vy of a particle in the string on which the wave travels. Express your answer in terms of the variables A,v,λ,x,t, and appropriate constants.
Find the maximum speed of a particle of the string. Express your answer in terms of the variables A,v,λ,x,t, and appropriate constants.

Answer & Explanation

Julie Mathew

Julie Mathew

Beginner2021-12-07Added 15 answers

Step 1
The general equation of the transverse wave travelling in a string is,
y(x,t)=Acos[2πλ(xvt)]
The velocity of a particle in the travelling wave is,
vy=v(x,t)
=dy(x,t)dt
Substitute Acos[2πλ(xvt)] for y(x,t) and differentiate to find the expression for the transverse velocity.
vy=ddt(Acos[2πλ(xvt)])
=2πvλAsin[2πλ(xvt)]
Part 1
The transverse velocity of a point in the string on which the wave travels is vy=2πvλAsin[2πλ(xvt)]
Step 2
The value of sine is bound between -1 and +1.
The maximum value of the velocity (vy)max, for a given frequency and wavelength, corresponds to the term
sin[2πλ(xvt)] taking the value +1.
Therefore,
vymax=2πvλA(1)
=2πvAλ
Part 2
The maximum transverse velocity of the particle in the string is vymax=2πvAλ.

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