To figure out potential and kinetic energy in simple harmonic motion (SHM), we can follow a few easy steps.

SHM happens in things like a mass on a spring or a pendulum swinging back and forth. In these systems, energy changes between potential and kinetic forms all the time.

1. Kinetic Energy (KE):
In SHM, kinetic energy is the highest when the object is in the middle of its movement, called the equilibrium position. The formula for kinetic energy is:

KE = 1/2 * m * v²

Here, m is the mass of the object, and v is how fast it’s going.

At the highest speed, you can replace v with a formula that uses the biggest distance from the center, called amplitude (A), and a special number called angular frequency (ω). This is written as:

v = A * ω * sin(ω * t)

So, you’ll want to look at how speed changes over time to find kinetic energy at different points during the motion.

2. Potential Energy (PE):
Potential energy in SHM is the highest when the object is at its maximum distance from the center, which is the amplitude (A). You can find the potential energy using this formula:

PE = 1/2 * k * x²

In this equation, k is the spring constant (for things like springs), and x is how far the object is from the center. The highest potential energy occurs when x equals the amplitude (A):

PE_max = 1/2 * k * A²

3. Energy Conservation in SHM:
One amazing thing about SHM is that the total energy stays the same. The total energy (E) in SHM is just the sum of kinetic and potential energy:

E = KE + PE

As energy changes between kinetic and potential forms, when the object is at its highest point, all of its energy is potential. But when it’s in the middle (equilibrium), all of its energy is kinetic. This shows how energy is conserved in action!

In real life, if you measure how far something is from the center, its mass, and the spring constant, you can easily calculate how the energy changes during SHM. It’s pretty cool to see how energy moves back and forth like this!