To analyze the energy in a (SHM) system, we begin by identifying all the energy forms it can possess. From Hooke’s Law, we know that a stretched or compressed spring stores , expressed as:

In the absence of energy loss (e.g., no friction), total mechanical energy is conserved. That means the total of and potential energy remains constant:

Or, in terms of variables:

This applies to any SHM system, including pendulums. For a simple pendulum, we use the substitutions:

• Velocity: [latex]v = L\omega[/latex]
• Spring constant: [latex]k = \frac{mg}{L}[/latex]
• Displacement: [latex]x = L\theta[/latex]

This gives us an energy expression for pendulums:

In an undamped SHM system, energy oscillates between kinetic and potential forms. For example, in a spring-mass system on a frictionless surface, as illustrated in Figure 120.1, the energy shifts from entirely potential (stored in the spring) to entirely kinetic (when passing equilibrium), and back again.

We can derive the velocity of an object in SHM using conservation of energy. Suppose the object starts from rest at its maximum displacement [latex]x = X[/latex], then the total energy is:

At any point in the motion:

Solving for [latex]v[/latex], we get:

This can be rewritten as:

Or in terms of maximum velocity:

Where:

Maximum velocity occurs at the equilibrium position ([latex]x = 0[/latex]). This velocity depends on:

• Amplitude: Larger amplitude [latex]X[/latex] increases [latex]v_{\text{max}}[/latex].
• Spring stiffness: Greater spring constant [latex]k[/latex] leads to higher [latex]v_{\text{max}}[/latex].
• Mass: Heavier objects ([latex]m[/latex]) reduce [latex]v_{\text{max}}[/latex] because acceleration is lower.

For a simple pendulum, we can derive a similar expression using angular displacement:

This relates the pendulum’s angular velocity to its maximum angular displacement [latex]\theta_{\text{max}}[/latex], length [latex]L[/latex], and gravitational acceleration [latex]g[/latex].