Harmonic oscillator

Calculate the total energy of a body performing a harmonic oscillating motion if its mass is 200 g, the amplitude of the deflection is 2 cm and the frequency is 5 Hz. Round the result to 3 decimal places.

Correct result:

E =  0.0395 J

Solution:

$$\begin{gathered} m=200\text{ g}\to \text{ kg}=200\mathrm{/}1000\text{ kg}=0.2\text{ kg } \\ x=2\text{ cm}\to \text{ m}=2\mathrm{/}100\text{ m}=0.02\text{ m } \\ f=5\text{ Hz } \\ g=9.81{\text{m/s}}^2 \\ T=1\mathrm{/}f=1\mathrm{/}5={\displaystyle \frac{1}{5}}=0.2\text{ s } \\ \omega =2\pi \cdot f=2\cdot 3.1416\cdot 5\doteq 31.41591\text{/rad } \\ f={\displaystyle \frac{1}{2\pi }}\cdot \sqrt{{\displaystyle \frac{k}{m}}} \\ k={\omega }^2\cdot m=31.4159^2\cdot 0.2\doteq 197.3921 \\ E={\displaystyle \frac{1}{2}}\cdot k\cdot x^2={\displaystyle \frac{1}{2}}\cdot 197.3921\cdot 0.02^2=0.0395\text{ J} \end{gathered}$$

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