Calculating Max Speed of a Particle on String

[vroman]

For a simple harmonic transverse wave that is propagating along a string toward the right, how is maximum speed of a particle on the string calculated?
Is it

4A/T
If A=amplitude and T=Period?

 

[geosonel]

For a simple harmonic transverse wave that is propagating along a string toward the right, how is maximum speed of a particle on the string calculated?
Is it

4A/T
If A=amplitude and T=Period?

displacement = s(x,t) = A*sin(ωt - kx + φ)
particle speed = v(x,t) = ∂s(x,t)/∂t = (ωA)*cos(ωt - kx + φ)
max particle speed (when cos=1) = ωA = 2πf*A = 2πA/T

 

[thepatientmental]

Yes, the maximum speed of a particle on a string can be calculated using the formula 4A/T, where A is the amplitude of the wave and T is the period of the wave. This formula is derived from the relationship between velocity, frequency, and wavelength in a wave, which is v = fλ. In a simple harmonic transverse wave, the frequency (f) is equal to 1/T, and the wavelength (λ) is equal to 2A. Therefore, the maximum velocity can be calculated as v = (1/T)(2A) = 2A/T. However, since the particle on a string undergoes two complete cycles (one full oscillation) in one period, the maximum speed is doubled to 4A/T. This formula can be used to calculate the maximum speed of any particle on a string for a given amplitude and period.