Final answer:
In simple harmonic motion, when the displacement is one-half of the amplitude, the kinetic energy constitutes three-fourths, or 75%, of the total energy. The correct answer to the student's question is option D. 3/4.
Step-by-step explanation:
In a system undergoing simple harmonic motion (SHM), the energy is divided between elastic potential energy (PE) and kinetic energy (KE), and the sum of these two forms of energy remains constant. When the displacement, x, is one-half the amplitude, A, the energy stored as elastic potential energy is given by PE = 1/2 k (0.5A)², where k is the spring constant. To find the fraction of energy that is kinetic when the displacement is half the amplitude, we can refer to the conservation of energy principle in SHM, which tells us that KE + PE = constant. At maximum displacement (amplitude, A), all the energy is potential, whereas at zero displacement (equilibrium), all the energy is kinetic.
Given the relationship for elastic potential energy, PE = 1/2 kx², and since x=A/2, we can substitute to find that PE = 1/2 k (A/2)² = 1/8 kA². The total energy in the system, when the object is at amplitude, would be PE = 1/2 kA², so the fraction of the kinetic energy at half amplitude is 1 - (PE at half amplitude / total energy). This gives us 1 - (1/8 kA²) / (1/2 kA²) = 1 - 1/4 = 3/4 or 75%. Therefore, when the displacement is one-half the amplitude in a simple harmonic oscillator, three-fourths, or 75%, of the total energy is kinetic energy.
The correct option that represents the fraction of the total energy that is kinetic when the displacement is half the amplitude in simple harmonic motion is D. 3/4.