Energy In Shm

Question

Total mechanical energy of SHM (mass m, amplitude A)?

RankGuru · Accepted Answer

1. **Potential Energy (U)**: $U = \frac{1}{2} k x^2$. At extreme position ($x=A$), $U_{max} = \frac{1}{2} k A^2$.
2. **Kinetic Energy (K)**: $K_{max} = \frac{1}{2} m v_{max}^2$ at $x=0$. Since $v_{max} = \omega A$, $K_{max} = \frac{1}{2} m (\omega A)^2$.
3. **Conservation**: Total Energy $E = U_{max} = K_{max} = \frac{1}{2} k A^2$.
4. **Substitute for k**: Since $\omega^2 = k/m$, then $k = m \omega^2$.
5. **Final Expression**: **$E = \frac{1}{2} m \omega^2 A^2$**.
6. **Insight**: Total energy is independent of $x$ and $t$, and proportional to the **square of the amplitude**.

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