Answer:
(a) f = 0.58Hz
(b) vmax = 0.364m/s
(c) amax = 1.32m/s^2
(d) E = 0.1J
(e) ![x(t)=0.1m\ cos(2\pi(0.58s^{-1})t)](https://tex.z-dn.net/?f=x%28t%29%3D0.1m%5C%20cos%282%5Cpi%280.58s%5E%7B-1%7D%29t%29)
Explanation:
(a) The frequency of the oscillation, in a spring-mass system, is calulated by using the following formula:
(1)
k: spring constant = 20.0N/m
m: mass = 1.5kg
you replace the values of m and k for getting f:
![f=\frac{1}{2\pi}\sqrt{\frac{20.0N/m}{1.5kg}}=0.58s^{-1}=0.58Hz](https://tex.z-dn.net/?f=f%3D%5Cfrac%7B1%7D%7B2%5Cpi%7D%5Csqrt%7B%5Cfrac%7B20.0N%2Fm%7D%7B1.5kg%7D%7D%3D0.58s%5E%7B-1%7D%3D0.58Hz)
The frequency of the oscillation is 0.58Hz
(b) The maximum speed is given by:
(2)
A: amplitude of the oscillations = 10.0cm = 0.10m
![v_{max}=2\pi (0.58s^{-1})(0.10m)=0.364\frac{m}{s}](https://tex.z-dn.net/?f=v_%7Bmax%7D%3D2%5Cpi%20%280.58s%5E%7B-1%7D%29%280.10m%29%3D0.364%5Cfrac%7Bm%7D%7Bs%7D)
The maximum speed of the mass is 0.364 m/s.
The maximum speed occurs when the mass passes trough the equilibrium point of the oscillation.
(c) The maximum acceleration is given by:
![a_{max}=\omega^2A=(2\pi f)^2 A](https://tex.z-dn.net/?f=a_%7Bmax%7D%3D%5Comega%5E2A%3D%282%5Cpi%20f%29%5E2%20A)
![a_{max}=(2\pi (0.58s^{-1}))(0.10m)=1.32\frac{m}{s^2}](https://tex.z-dn.net/?f=a_%7Bmax%7D%3D%282%5Cpi%20%280.58s%5E%7B-1%7D%29%29%280.10m%29%3D1.32%5Cfrac%7Bm%7D%7Bs%5E2%7D)
The maximum acceleration is 1.32 m/s^2
The maximum acceleration occurs where the elastic force is a maximum, that is, where the mass is at the maximum distance from the equilibrium point, that is, the acceleration.
(d) The total energy of the system is:
![E=\frac{1}{2}kA^2=\frac{1}{2}(20.0N/m)(0.10m)^2=0.1J](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B1%7D%7B2%7DkA%5E2%3D%5Cfrac%7B1%7D%7B2%7D%2820.0N%2Fm%29%280.10m%29%5E2%3D0.1J)
The total energy is 0.1J
(e) The displacement as a function of time is:
![x(t)=Acos(\omega t)=Acos(2\pi ft)\\\\x(t)=0.1m\ cos(2\pi(0.58s^{-1})t)](https://tex.z-dn.net/?f=x%28t%29%3DAcos%28%5Comega%20t%29%3DAcos%282%5Cpi%20ft%29%5C%5C%5C%5Cx%28t%29%3D0.1m%5C%20cos%282%5Cpi%280.58s%5E%7B-1%7D%29t%29)