Energy in SHM and Damped Oscillations

TL;DR

In Simple Harmonic Motion (SHM), mechanical energy constantly converts between kinetic and potential forms but its total remains constant. Damping introduces a resistive force that removes mechanical energy from the system over time, causing the oscillation's amplitude to decrease. The type of damping depends on how quickly this energy loss occurs.

1. The Mental Model

Imagine a perfect pendulum swinging forever; its total energy never changes, just shifts between speed (kinetic) and height (potential). Now imagine that same pendulum swinging in treacle; it slows down and stops because the treacle sucks energy out of it.

2. The Core Material

Energy in SHM

Photo by Ann H on Pexels

In ideal SHM, there's no friction or air resistance, so the total mechanical energy (TME) of the system stays constant. This TME is the sum of its kinetic energy (KE) and potential energy (PE).

• Kinetic Energy (KE): This is the energy of motion. It's maximum when the oscillating object is passing through its equilibrium position (where its speed is highest) and zero at the extreme ends of its swing (where it momentarily stops).
  • $KE = \frac{1}{2}mv^2$
• Potential Energy (PE): This is stored energy due to the object's position or state. For a spring-mass system, it's elastic potential energy; for a pendulum, it's gravitational potential energy. It's maximum at the extreme ends of the swing (where the spring is most stretched/compressed or the pendulum is highest) and zero at the equilibrium position.
  • For a spring: $PE = \frac{1}{2}kx^2$ (where $k$ is the spring constant and $x$ is displacement)

The fascinating part is how these two energies swap back and forth. When KE is max, PE is min, and vice-versa. Their sum, the TME, is constant and proportional to the square of the amplitude ($A^2$).

graph TD
    A["Equilibrium Position"] --> B{"Max KE / Min PE"}
    B --> C["Moving Towards Extreme"]
    C --> D{"KE decreasing / PE increasing"}
    D --> E["Extreme Position"]
    E --> F{"Min KE / Max PE"}
    F --> G["Moving Towards Equilibrium"]
    G --> H{"KE increasing / PE decreasing"}
    H --> A;

Damped Oscillations

Photo by Egor Komarov on Pexels

Real-world oscillations don't last forever because energy is always lost to the surroundings, usually as heat or sound. This energy loss is called damping, and it causes the amplitude of the oscillation to decrease over time.

The resistive force (damping force) is often proportional to the velocity of the oscillating object. Think of an object moving through a fluid like air or water – the faster it moves, the more resistance it experiences.

There are three main types of damping:

1. Underdamped: This is what we typically imagine when we think of a damped oscillation. The system still oscillates, but its amplitude gradually decreases over time until it eventually comes to rest. The energy is slowly removed.
2. Critically Damped: This is the quickest way for a system to return to its equilibrium position without oscillating at all. There's just enough damping to stop any overshoots. Think of a door closer that slowly and smoothly shuts a door without letting it slam or bounce.
3. Overdamped: The system returns to equilibrium slowly without oscillating, but it takes longer than critical damping. It moves sluggishly towards equilibrium, like a door closing very slowly in thick treacle.

The rate of energy loss determines the type of damping. In underdamped systems, energy decreases exponentially over time.

3. Worked Example

Let's consider a spring-mass system with a mass of $0.5 \text{ kg}$ oscillating horizontally. The spring constant is $20 \text{ N/m}$, and the amplitude of oscillation is $0.1 \text{ m}$.

a) Calculate the total mechanical energy in ideal SHM.

The total mechanical energy (TME) in SHM is constant. We can calculate it at the point of maximum displacement (the amplitude), where all the energy is potential energy:

$TME = PE_{max} = \frac{1}{2}kA^2$
$TME = \frac{1}{2} (20 \text{ N/m}) (0.1 \text{ m})^2$
$TME = \frac{1}{2} (20) (0.01)$
$TME = 0.1 \text{ J}$

So, the total mechanical energy of the system is $0.1 \text{ J}$. This energy will constantly swap between kinetic and potential forms but always sum to $0.1 \text{ J}$.

b) If the system becomes underdamped, what happens to this $0.1 \text{ J}$ over time?

If the system is underdamped, the initial total mechanical energy of $0.1 \text{ J}$ will gradually decrease over time. The damping force will do negative work on the system, converting this mechanical energy into other forms, primarily heat due to friction or air resistance. The amplitude of oscillation will get smaller and smaller, and eventually, the system will come to rest, at which point its mechanical energy will be zero.

4. Key Takeaways

• In ideal SHM, total mechanical energy (kinetic + potential) is always conserved.
• Energy constantly converts between kinetic (maximum at equilibrium) and potential (maximum at extremes).
• Total mechanical energy in SHM is proportional to the square of the amplitude ($A^2$).
• Damping is the loss of mechanical energy from an oscillating system, usually due to resistive forces.
• Damping causes the amplitude of oscillations to decrease over time.
• Underdamped systems oscillate with decreasing amplitude.
• Critically damped systems return to equilibrium fastest without oscillating.
• Overdamped systems return to equilibrium slowly without oscillating.

Common Mistakes to Avoid:
- Assuming total energy is always conserved in any oscillation; it's only true for ideal SHM.
- Confusing the point of maximum KE with maximum PE; they occur at different positions.
- Thinking damping stops oscillations immediately; underdamped systems still oscillate.
- Believing critical damping means no movement; it means the fastest return to rest without overshooting.

5. Now Try It

Imagine a child on a swing. Describe the energy transformations (KE and PE) as they swing from their highest point, through the lowest point, and back up to the other highest point. Then, consider what happens to the swing's motion and energy if someone lightly pushes them every swing (driving force) or if there's significant air resistance (damping). What would happen if the air resistance was very high, like swinging in thick mud?