The potential energy of a simple harmonic oscillator when the particle is half way to its end point is

The potential energy of a simple harmonic oscillator when the particle is halfway to its endpoint is:

Detailed Answer:

The total energy of a simple harmonic oscillator is given by: \[ E_{\text{total}} = \frac{1}{2} k A^2 \] where: - \( k \) is the spring constant, - \( A \) is the amplitude of oscillation. The potential energy at any displacement \( x \) is: \[ U = \frac{1}{2} k x^2 \] When the particle is halfway to the endpoint: \[ x = \frac{A}{2} \] Substituting \( x = \frac{A}{2} \) into the potential energy equation: \[ U = \frac{1}{2} k \left(\frac{A}{2}\right)^2 \] \[ U = \frac{1}{2} k \frac{A^2}{4} \] \[ U = \frac{1}{8} k A^2 \] The total energy is \( E_{\text{total}} = \frac{1}{2} k A^2 \), so the fraction of potential energy is: \[ \frac{U}{E_{\text{total}}} = \frac{\frac{1}{8} k A^2}{\frac{1}{2} k A^2} = \frac{1}{4} \] Therefore, the potential energy when the particle is halfway to its endpoint is: \[ \boxed{\frac{1}{4} \, E_{\text{total}}}. \]