In simple harmonic motion, at which position is the kinetic energy equal to the potential energy for an amplitude A?

In simple harmonic motion (SHM), the total mechanical energy is conserved and is a combination of kinetic energy (KE) and potential energy (PE). The potential energy is maximum at the maximum displacement (amplitude) and zero at the equilibrium position, while kinetic energy is maximum at the equilibrium position and zero at the maximum displacement.

At a given position during SHM, the relationship between kinetic energy and potential energy can be described by the formulas:

• Potential Energy (PE) = (1/2) k x²

• Kinetic Energy (KE) = (1/2) k (A² - x²)

where k is the spring constant and x is the displacement from the equilibrium position.

For the situation where kinetic energy is equal to potential energy (KE = PE), we set the two equations equal to each other:

(1/2) k x² = (1/2) k (A² - x²)

Dividing both sides by (1/2) k (as long as k is not zero), we have:

x² = A² - x²

This rearranges to:

2x² = A²

From here, we can solve for x:

x² = A² /