🎢 Energy in Simple Harmonic Motion (SHM): The Physics Rollercoaster

Welcome to one of the most elegant parts of A-Level Physics! In the previous chapter, we explored the kinematics of Simple Harmonic Motion (SHM)—how things move. Now, we dive into the "why"—the forces and, crucially, the energy that drives this continuous, beautiful oscillation.

Understanding energy in SHM is key to solving tough exam problems, as it connects motion (kinematics) directly to mass and amplitude. Don't worry if the formulas look intimidating; at its heart, this topic is just about a continuous, perfect energy swap!

Key Learning Outcomes (Syllabus 17.2)

  • Describe the continuous interchange between kinetic and potential energy.
  • Recall and use the formula for the total energy of an SHM system.

1. The Great Energy Swap: Kinetic and Potential Energy Interchange

In any system undergoing ideal Simple Harmonic Motion (meaning we ignore friction and air resistance), the total mechanical energy remains constant. However, this energy is constantly being exchanged between two forms:

1. Kinetic Energy (\(E_K\)): The energy of motion.
2. Potential Energy (\(E_P\)): The stored energy due to the system's position (e.g., gravitational PE for a pendulum, or elastic PE for a mass on a spring).

Analogy: The Perfect Swing

Imagine a child on a perfect, frictionless swing.

  • At the peak (Maximum Displacement, \(x = \pm x_0\)): The swing momentarily stops. All the energy is stored. It has maximum Potential Energy (\(E_{P, max}\)) and zero Kinetic Energy (\(E_{K} = 0\)).
  • At the bottom (Equilibrium Position, \(x = 0\)): The swing is moving at its fastest speed. All the stored energy has been converted to motion. It has maximum Kinetic Energy (\(E_{K, max}\)) and zero Potential Energy (\(E_{P} = 0\)).

The total energy is simply the sum of these two energies at any point: $$ E_{Total} = E_K + E_P $$

Important Note: This energy swap happens continuously, twice every complete cycle.

Quick Review: Key Energy Points
Position Displacement (\(x\)) Kinetic Energy (\(E_K\)) Potential Energy (\(E_P\))
Equilibrium \(x = 0\) Maximum Zero
Amplitude \(x = \pm x_0\) Zero Maximum

2. Calculating Kinetic Energy (\(E_K\))

We know that kinetic energy is \(E_K = \frac{1}{2}mv^2\).

In SHM, the velocity \(v\) changes with displacement \(x\) according to the equation: $$ v = \pm \omega \sqrt{x_0^2 - x^2} $$

If we substitute this definition of \(v\) into the kinetic energy equation, we get the instantaneous kinetic energy at any displacement \(x\): $$ E_K = \frac{1}{2} m \left( \omega \sqrt{x_0^2 - x^2} \right)^2 $$ $$ E_K = \frac{1}{2} m \omega^2 (x_0^2 - x^2) $$

Step-by-Step for Maximum KE:
The kinetic energy is maximum when the displacement \(x = 0\) (at the equilibrium position).
1. Set \(x = 0\): $$ E_{K, max} = \frac{1}{2} m \omega^2 (x_0^2 - 0) $$ 2. Result: $$ E_{K, max} = \frac{1}{2} m \omega^2 x_0^2 $$

Key Takeaway: Maximum KE depends on mass, angular frequency, and the square of the amplitude.

3. Calculating Potential Energy (\(E_P\))

Potential energy in SHM is the energy stored due to the object's displacement from equilibrium. While the specific formula might differ (springs use elastic PE, pendulums use gravitational PE), in SHM, it is always related to the displacement \(x\).

Since the total energy is conserved (\(E_{Total} = E_K + E_P\)), we can find the potential energy at any point by subtracting the kinetic energy from the total energy.

$$ E_P = E_{Total} - E_K $$

Substituting the full expressions for \(E_{Total}\) (which we established equals \(E_{K, max}\) in the section above) and \(E_K\):

$$ E_P = \left( \frac{1}{2} m \omega^2 x_0^2 \right) - \left( \frac{1}{2} m \omega^2 x_0^2 - \frac{1}{2} m \omega^2 x^2 \right) $$

Simplifying this (the \(\frac{1}{2} m \omega^2 x_0^2\) terms cancel out): $$ E_P = \frac{1}{2} m \omega^2 x^2 $$

Step-by-Step for Maximum PE:
The potential energy is maximum when the displacement \(x\) is maximum (i.e., \(x\) is equal to the amplitude \(x_0\)).
1. Set \(x = x_0\): $$ E_{P, max} = \frac{1}{2} m \omega^2 x_0^2 $$

Key Takeaway: The formula for Potential Energy (\(E_P\)) shows that it is always zero at equilibrium (\(x=0\)) and maximized at the amplitude (\(x=x_0\)).

4. The Total Energy Equation for SHM

This is the central result required by the syllabus (17.2.2). Because energy is conserved, the Total Energy (\(E\)) of the system is equal to the maximum KE (or maximum PE).

The total energy of a system undergoing simple harmonic motion is given by:

$$ \mathbf{E = \frac{1}{2} m \omega^2 x_0^2} $$

Memory Aid: Notice how this equation strongly resembles the kinetic energy formula, \(E_K = \frac{1}{2}mv^2\). Here, we have replaced the maximum velocity squared (\(v_{max}^2\)) with its SHM equivalent: \((\omega x_0)^2\).

Did you know? Connection to Spring Constant (k)

For a mass-spring system, recall the angular frequency formula: \(\omega^2 = k/m\).
If we substitute this into the total energy equation: $$ E = \frac{1}{2} m \left( \frac{k}{m} \right) x_0^2 $$ $$ E = \frac{1}{2} k x_0^2 $$

This is the standard formula for Elastic Potential Energy (where \(x_0\) is the maximum extension). This confirms that the total mechanical energy in SHM is exactly the maximum potential energy stored by the spring!

Common Mistake to Avoid

Do not confuse the instantaneous displacement \(x\) with the amplitude \(x_0\).

  • \(x_0\) is the constant maximum displacement, used to calculate the Total Energy (\(E_{Total}\)).
  • \(x\) is the variable displacement at any moment, used to calculate the instantaneous Potential Energy (\(E_P\)) or Kinetic Energy (\(E_K\)).

5. Graphical Analysis of Energy in SHM

Interpreting the energy graphs helps reinforce the concept of interchange. If we plot Kinetic Energy (\(E_K\)) and Potential Energy (\(E_P\)) against the displacement \(x\), we see parabolic curves.

Both kinetic energy and potential energy graphs are always positive because they depend on the square of displacement (\(x^2\)) or the square of velocity (\(v^2\)). Energy is a scalar quantity.

Potential Energy vs Displacement Graph

Since \(E_P \propto x^2\), the graph is a parabola that opens upwards, with its minimum at equilibrium \(x=0\) and maxima at the amplitudes \(x=\pm x_0\).

Kinetic Energy vs Displacement Graph

Since \(E_K \propto (x_0^2 - x^2)\), the graph is a parabola that opens downwards, with its maximum at equilibrium \(x=0\) and zeroes at the amplitudes \(x=\pm x_0\).

The sum of the two graphs, \(E_K + E_P\), should result in a flat horizontal line representing the constant Total Energy (\(E_{Total}\)).

This graphical relationship shows clearly that when one energy reaches zero, the other reaches its maximum, maintaining a constant total energy throughout the motion.


🌟 Chapter Summary: Energy in SHM 🌟

The energy in SHM is defined by a beautiful, constant transformation between kinetic and potential energy.

  • Interchange: KE is maximum at equilibrium (\(x=0\)), while PE is maximum at the amplitude (\(x=\pm x_0\)).
  • Conservation: The total mechanical energy \(E\) remains constant throughout the oscillation (assuming no damping).
  • Key Formula (Total Energy): Use the angular frequency and amplitude to find the total energy: $$ E = \frac{1}{2} m \omega^2 x_0^2 $$