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Step-by-Step Solution
Step 1: Express the Known Kinetic Energy
The kinetic energy (K.E.) of the particle in simple harmonic motion (SHM) is given as
$K = K_{0}\cos^{2}\bigl(\omega t\bigr)$.
Note that $K_{0}$ is a constant representing the maximum value of the kinetic energy when $\cos^{2}(\omega t) = 1$.
Step 2: State the Total Energy in SHM
For a particle undergoing SHM, the total energy (T.E.) is the sum of kinetic energy (K.E.) and potential energy (P.E.). Let the total energy be $E$. Hence,
$E = K + U,$
where $U$ denotes the potential energy.
Step 3: Identify Maximum Kinetic Energy
The maximum value of $K = K_{0}\cos^{2}\bigl(\omega t\bigr)$ occurs when $\cos^{2}(\omega t) = 1$. In that case,
$K_{\text{max}} = K_{0}.$
Step 4: Determine the Total Energy
At the instant when kinetic energy is maximum, the potential energy is zero (because for SHM at the mean position, displacement is zero, and hence potential energy is zero). Therefore,
$E = K_{\text{max}} + 0 = K_{0}.$
Thus, the total energy $E$ is $K_{0}$.
Step 5: Find the Maximum Potential Energy
At the extreme position, the displacement is maximum and kinetic energy is zero, implying that all the energy is stored as potential energy. Therefore,
$\text{Maximum P.E.} = E = K_{0}.$
Step 6: State the Final Answer
The maximum potential energy is $K_{0}$ and the total energy is also $K_{0}$.
