Answer:
[tex]x(t) = 8\cdot \cos 9.798\cdot t[/tex]
Step-by-step explanation:
The equation of motion for the spring-mass system is:
[tex]x(t) = A\cdot \cos (\omega\cdot t + \phi)[/tex]
Where:
[tex]\omega = \sqrt{\frac{k}{m}}[/tex]
[tex]k[/tex] - Spring constant, in [tex]\frac{lbm}{s^{2}}[/tex].
[tex]m[/tex] - Mass, in [tex]lbm[/tex].
The spring constant is:
[tex]k = \frac{m\cdot g}{\Delta s}[/tex]
[tex]k = \frac{(24\,lbm)\cdot (32\,\frac{ft}{s^{2}} )}{\frac{4}{12} \,ft}[/tex]
[tex]k = 2304\,\frac{lbm}{s^{2}}[/tex]
The angular frequency is:
[tex]\omega = \sqrt{\frac{2304\,\frac{lbm}{s^{2}} }{24\,lbm} }[/tex]
[tex]\omega \approx 9.798\,\frac{rad}{s}[/tex]
The initial condition for the system is:
[tex]x(0) = +8\,in[/tex]
[tex]v(0) = 0\,\frac{in}{s}[/tex]
The function for speed is obtained by deriving the previous function:
[tex]v(t) = -\omega \cdot A\cdot \sin (\omega\cdot t + \phi)[/tex]
The following expressions are formed by substituting all known variables:
[tex]A \cdot \cos \phi = 8\,in[/tex]
[tex]-(9.798\,\frac{rad}{s} )\cdot A \cdot \sin \phi = 0\,\frac{in}{s}[/tex]
The phase angle is found by dividing the initial velocity by the initial position:
[tex]\tan \phi = 0[/tex]
[tex]\phi = 0\,rad[/tex]
The amplitude is:
[tex]A = 8\,in[/tex]
The equation of motion is:
[tex]x(t) = 8\cdot \cos 9.798\cdot t[/tex]
