Answer:
The maximum acceleration of the system is 359.970 centimeters per square second.
Explanation:
The motion of the mass-spring system is represented by the following formula:
![x(t) = A\cdot \cos (\omega \cdot t + \phi)](https://tex.z-dn.net/?f=x%28t%29%20%3D%20A%5Ccdot%20%5Ccos%20%28%5Comega%20%5Ccdot%20t%20%2B%20%5Cphi%29)
Where:
- Position of the mass with respect to the equilibrium position, measured in centimeters.
- Amplitude of the mass-spring system, measured in centimeters.
- Angular frequency, measured in radians per second.
- Time, measured in seconds.
- Phase, measured in radians.
The acceleration experimented by the mass is obtained by deriving the position equation twice:
![a (t) = -\omega^{2}\cdot A \cdot \cos (\omega\cdot t + \phi)](https://tex.z-dn.net/?f=a%20%28t%29%20%3D%20-%5Comega%5E%7B2%7D%5Ccdot%20A%20%5Ccdot%20%5Ccos%20%28%5Comega%5Ccdot%20t%20%2B%20%5Cphi%29)
Where the maximum acceleration of the system is represented by
.
The natural frequency of the mass-spring system is:
![\omega = \sqrt{\frac{k}{m} }](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Csqrt%7B%5Cfrac%7Bk%7D%7Bm%7D%20%7D)
Where:
- Spring constant, measured in newtons per meter.
- Mass, measured in kilograms.
If
and
, the natural frequency is:
![\omega = \sqrt{\frac{12\,\frac{N}{m} }{0.40\,kg} }](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Csqrt%7B%5Cfrac%7B12%5C%2C%5Cfrac%7BN%7D%7Bm%7D%20%7D%7B0.40%5C%2Ckg%7D%20%7D)
![\omega \approx 5.477\,\frac{rad}{s}](https://tex.z-dn.net/?f=%5Comega%20%5Capprox%205.477%5C%2C%5Cfrac%7Brad%7D%7Bs%7D)
Lastly, the maximum acceleration of the system is:
![a_{max} = \left(5.477\,\frac{rad}{s})^{2}\cdot (12\,cm)](https://tex.z-dn.net/?f=a_%7Bmax%7D%20%3D%20%5Cleft%285.477%5C%2C%5Cfrac%7Brad%7D%7Bs%7D%29%5E%7B2%7D%5Ccdot%20%2812%5C%2Ccm%29)
![a_{max} = 359.970\,\frac{cm}{s^{2}}](https://tex.z-dn.net/?f=a_%7Bmax%7D%20%3D%20359.970%5C%2C%5Cfrac%7Bcm%7D%7Bs%5E%7B2%7D%7D)
The maximum acceleration of the system is 359.970 centimeters per square second.