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,
- Time period ([tex]T[/tex]) = [tex]2s[/tex]
- [tex]A=8[/tex]
- 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]
