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A buoy floating in the sea is bobbing in simple harmonic motion with period 2 seconds and amplitude 8 in. Its displacement d from sea level at time t=0 seconds is 0 in, and initially it moves downward. (Note that downward is the negative direction.) Give the equation modeling the displacement d as a function of time t. d= ク

Respuesta :

Answer:

The equation of the displacement [tex]d[/tex] as a function of time [tex]t[/tex] is :

[tex]d(t)=8sin(\pi t+\pi )[/tex]

Step-by-step explanation:

Generally , A simple harmonic wave is a sinusoidal function that is it can be expressed in simple [tex]sin[/tex] or [tex]cos[/tex] terms.

Thus,

[tex]d(t) = Asin(wt+c)[/tex]

is the general form of displacement of a SHM.

where,

  • [tex]d(t)[/tex] is the displacement with respect to the mean position at any time [tex]t[/tex]
  • [tex]A[/tex] is amplitude
  • [tex]w[/tex] is the natural frequency of oscillation ([tex]rads^{-1}[/tex])
  • [tex]c[/tex] is the phase angle which indicates the initial position of the object in SHM ([tex]rad[/tex])

given,

  1. Time period ([tex]T[/tex]) = [tex]2s[/tex]
  2. [tex]A=8[/tex]
  3. The natural frequency ([tex]w[/tex]) and time period ([tex]T[/tex]) is :

                               [tex]w=\frac{2\pi} {T}[/tex]

∴

[tex]w[/tex] = [tex]\frac{2\pi }{2}  = \pi[/tex] [tex]rads^{-1}[/tex]

∴

the equation :

⇒[tex]d(t)=8sin(\pi t+c)[/tex]                        ------1

since [tex]d=0[/tex] when [tex]t=o[/tex] ,

⇒[tex]0=8sinc\\c=n\pi[/tex]                         ------2

where n is an integer ;

⇒since the bouy immediately moves in the negative direction , [tex]x[/tex] must be negative or c must be an odd multiple of [tex]\pi[/tex].

⇒ [tex]d(t) = 8sin(\pi t+(2k+1)\pi )[/tex]         ------3

where k is also an integer ;

the least value of [tex]k=0[/tex];

thus ,

the equation is :

[tex]d(t)=8sin(\pi t+\pi )[/tex]