In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is kinetic?

In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is kinetic?

1. 1/2
2. 3/4
3. zero
4. 1/4

Detailed Answer:

The total energy \( E \) in simple harmonic motion (SHM) is constant and given by: \[ E = \frac{1}{2} k A^2 \] where: - \( k \) is the spring constant, - \( A \) is the amplitude. The energy is divided into potential energy (P.E.) and kinetic energy (K.E.). At any displacement \( x \): \[ \text{P.E.} = \frac{1}{2} k x^2, \quad \text{K.E.} = E - \text{P.E.} \] When the displacement \( x = \frac{A}{2} \), the potential energy becomes: \[ \text{P.E.} = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{4} E \] Therefore, the kinetic energy is: \[ \text{K.E.} = E - \text{P.E.} = E - \frac{1}{4} E = \frac{3}{4} E \] Thus, the fraction of the total energy that is kinetic is: \[ \boxed{\frac{3}{4}} \] Correct answer: (b) 3/4.