Oscillations

The First Law of Thermodynamics & Oscillations

What is the First Law?

The First Law is a statement of energy conservation for thermodynamic systems:

\$\Delta U = Q - W\$

where

  • ΔU = change in internal energy

  • Q = heat added to the system (positive if added)

  • W = work done by the system (positive if the system does work)

Think of it like a bank account: money (energy) can be deposited (heat) or withdrawn (work), but the total balance changes accordingly.

Connecting to Oscillations

Oscillatory systems (springs, pendulums, LC circuits) exchange energy between two forms: kinetic and potential.

When no external heat or work is added, the total mechanical energy remains constant, mirroring the First Law with \(Q=W=0\).


🔄 Example: A mass on a spring oscillates, converting spring potential energy to kinetic energy and back, just like a pendulum swings back and forth.

Energy Flow in a Simple Spring Oscillator

TimePotential Energy (U)Kinetic Energy (K)Total Energy (E)
0max0constant
T/4halfhalfconstant
T/20maxconstant

Real‑World Analogy

Imagine a playground swing.

  • When you push (doing work), you add energy (Q).

  • As the swing moves, that energy shifts between potential (height) and kinetic (speed).

  • If you let go and no friction acts, the swing keeps oscillating forever – total energy stays the same.

This is a perfect illustration of the First Law: energy is neither created nor destroyed, just transformed.

Key Takeaways for A‑Level

  1. Write the First Law as \( \Delta U = Q - W \) and remember the sign conventions.

  2. For isolated oscillatory systems, \(Q=W=0\) → \( \Delta U = 0 \) → total mechanical energy is constant.

  3. Use energy conservation to solve for unknowns in simple harmonic motion problems.

  4. Always check units: joules for energy, watts for power, etc.

Quick Practice Question

A 0.5 kg mass is attached to a spring with \(k = 200\,\text{N/m}\).

It is displaced 0.1 m from equilibrium and released from rest.


What is the maximum kinetic energy of the mass during oscillation?


🧠 Hint: Use \(E_{\text{max}} = \frac{1}{2}kA^2\).


Answer: \(E_{\text{max}} = \frac{1}{2} \times 200 \times 0.1^2 = 1\,\text{J}\).