A particle executing simple harmonic motion has a kinetic energy of $K = K_0 \cos^2(\omega t)$. The values of

Question

A particle executing simple harmonic motion has a kinetic energy of $K = K_0 \cos^2(\omega t)$. The values of the maximum potential energy and the total energy are, respectively:

Accepted Answer

1. **Analyze Kinetic Energy:** The given kinetic energy is $K = K_0 \cos^2(\omega t)$. The maximum value of $\cos^2(\omega t)$ is 1. Therefore, the maximum kinetic energy is $K_{max} = K_0$.
2. **Total Energy ($E$):** In simple harmonic motion, the total mechanical energy is conserved and is equal to the maximum kinetic energy (at the mean position) or the maximum potential energy (at the extreme position) . Thus, Total Energy $E = K_{max} = K_0$.
3. **Maximum Potential Energy ($U_{max}$):** Since the total energy is the sum of kinetic and potential energies ($E = K + U$), the potential energy is maximum when the kinetic energy is minimum (zero). At this point, the entire energy is potential. Therefore, $U_{max} = E = K_0$.
4. **Conclusion:** Both the maximum potential energy and the total energy are equal to $K_0$.

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