Answer:
See attached pictures.
Explanation:
See attachments for explanation.
[tex]y[/tex] term against the [tex]t[/tex] can be calculated by the damped vibration equation.[tex]y[/tex] term against the [tex]t[/tex] is,
[tex]y(t) = \dfrac 9{16} e^{-8t} - \dfrac 9{16} e^{-24t}[/tex]
The force applied to an object is directly proportional to the mass and acceleration of the object.
[tex]F = mg[/tex]
F = force= 4 pounds
m - mass = ?
g - gravitaional acceleration =[tex]\bold {32\ ft/s^2}[/tex]
So,
[tex]m= \dfrac 4{32}\\\\m = \dfrac 18[/tex]
Form Hook's law,
[tex]F= kx[/tex]
Where,
[tex]x[/tex]- the distance of stretch = 1/6 ft,
Do,
[tex]4 = k (\dfrac 16)\\k = 24\rm \ lb/ft[/tex]
As given in the question,
The air resistance is 4 times greater than the velocity,
So, the damped vibration of the spring-mass system,
[tex]m \dfrac {d^2y}{dt^2}+ c \dfrac {dy}{dt} + ky = 0[/tex]
Put the values in the formula,
[tex](\dfrac 18) \dfrac {d^2y}{dt^2}+ (4) \dfrac {dy}{dt} + (24)y = 0\\\\\dfrac {d^2y}{dt^2}+ (32) \dfrac {dy}{dt} + (192)y = 0[/tex]
Thus, [tex]y[/tex] term against the [tex]t[/tex] is,
[tex]y(t) = \dfrac 9{16} e^{-8t} - \dfrac 9{16} e^{-24t}[/tex]
To know more about damped vibration,
https://brainly.com/question/7229759
