Question

If one holds one end of the clothesline taut and wiggles it up and down sinusoidally with frequency 2.00 Hz, the amplitude is 0.075 m. The wave speed on the clothesline is 12 m/s. At , we have maximum positive displacement at position and the velocity is zero. Assume that no wave bounces back from the far end. (a) Find the wave amplitude , angular frequency , period , wavelength and wave number . (b) Find a wave function describing the wave. (c) Find equations for the displacement, as a function of time, of the particle at and of a point at 3.00 m.

Answer 1

a)

The wave amplitude A = 0.075 m

Angular frequency = 12.57 rad/s

Period = 0.5 s

Wavelength = 6.0 m

Wave number = 1.0475 m⁻¹

b)

The wave function describing the wave =
`y(x,t) = 0.075 m \ cos \ [(1.0475 \ rad/m)x - (12.57 \ rad/s) t]`

c) y= 0.0748875 + 0.9760 t

Step-by-step explanation:

a)

The wave amplitude A = 0.075 m

Angular frequency (ω) = 2πf

= 2× π× 2

= 12.57 rad/s

Period
`T = (1)/(f)`

`T = (1)/(2)\\\\T= 0.5 \ s`

Wavelength
`\lambda = (v)/(f)`

`\lambda = (12)/(2) \\\\\lambda = 6.0 \ m`

Wave number
`k = (\omega )/(v)`

`k = (12.57)/(12)\\\\k = 1.0475 \ m^(-1)`

b)

The wave equation describing the wave can be illustrated as:

`y(x,t) = A \ cos \ (kx - \omega t)`

`y(x,t) = 0.075 m \ cos \ [(1.0475 \ rad/m)x - (12.57 \ rad/s) t]`

c)

The equation for the displacement as a function of time of the particle at and of a point at 3.00 m is as follows:

`y= 0.075 \ cos \ [(1.0475 *3) - (12.57 ) t]`

`y= 0.075 *0.9985+ 0.9760 \ t\\\\y = 0.0748875 + 0.9760 t`