If a simple harmonic oscillator has a total energy E, what is the kinetic energy

If a simple harmonic oscillator has a total energy E, what is the kinetic energy when the displacement is half of the amplitude?

If a simple harmonic oscillator has a total energy E, what is the kinetic energy when the displacement is half of the amplitude?

Step 1: Understand that a simple harmonic oscillator has total energy E, which is the sum of kinetic energy (KE) and potential energy (PE).
Step 2: Recognize that when the displacement (x) is half of the amplitude (A), we can express this as x = A/2.
Step 3: Use the formula for potential energy in a simple harmonic oscillator, which is PE = (1/2)kx^2, where k is the spring constant.
Step 4: Substitute x = A/2 into the potential energy formula: PE = (1/2)k(A/2)^2 = (1/2)k(A^2/4) = (1/8)kA^2.
Step 5: The total energy E of the oscillator is given by E = (1/2)kA^2.
Step 6: Now, we can find the potential energy when x = A/2: PE = (1/8)kA^2.
Step 7: Since total energy E is conserved, we can find the kinetic energy (KE) using the equation: KE = E - PE.
Step 8: Substitute the values: KE = E - (1/8)kA^2.
Step 9: Since E = (1/2)kA^2, we can rewrite KE as KE = (1/2)kA^2 - (1/8)kA^2.
Step 10: Simplify the expression: KE = (4/8)kA^2 - (1/8)kA^2 = (3/8)kA^2.
Step 11: Now, we can express KE in terms of total energy E: Since E = (1/2)kA^2, we can say KE = (3/4)E.

Simple Harmonic Motion – Understanding the relationship between total energy, kinetic energy, and potential energy in a simple harmonic oscillator.
Energy Conservation – Applying the principle of conservation of energy to determine kinetic and potential energy at different displacements.
Amplitude and Displacement – Recognizing how displacement relates to amplitude in the context of simple harmonic motion.