Oscillations

The First Law of Thermodynamics

The first law expresses the principle of conservation of energy for thermodynamic systems. It states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system on its surroundings.

$$\Delta U = Q - W$$

Key Terms

• Internal energy ($U$): total microscopic kinetic and potential energy of the particles in the system.
• Heat ($Q$): energy transferred because of a temperature difference.
• Work ($W$): energy transferred when a force acts through a distance (e.g., $W = \int p\,dV$ for pressure–volume work).

Mathematical Formulation

For an infinitesimal process the first law can be written as

$$dU = \delta Q - \delta W$$

In many A‑Level problems the work term is expressed as $p\,dV$, giving

$$dU = \delta Q - p\,dV$$

Application to Oscillating Systems

Oscillatory motion (e.g., a mass‑spring system) involves continual inter‑conversion between kinetic and potential energy. When the oscillator is coupled to a thermal reservoir, the first law governs the energy exchange.

Energy in a Simple Harmonic Oscillator (SHO)

Quantity	Expression
Kinetic Energy ($K$)	$K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t)$
Elastic Potential Energy ($U_s$)	$U_s = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega t)$
Total Mechanical Energy ($E_{\text{mech}}$)	$E_{\text{mech}} = K + U_s = \frac{1}{2} k A^2$ (constant if no dissipation)

When damping is present (e.g., due to air resistance), mechanical energy is not conserved. The lost mechanical energy appears as heat transferred to the surroundings, and the first law becomes

$$\Delta U_{\text{int}} = Q_{\text{diss}} = \int F_{\text{damp}}\,dx$$

Damped Oscillator Example

1. Write the equation of motion with a damping force $F_{\text{damp}} = -b v$.
2. Show that the mechanical energy decays exponentially: $E_{\text{mech}}(t) = E_0 e^{-bt/m}$.
3. Identify the heat generated: $Q_{\text{diss}} = E_0 - E_{\text{mech}}(t)$.
4. Apply the first law: $\Delta U_{\text{int}} = Q_{\text{diss}}$ (no external work).

Thermal Driving of Oscillations

In some systems a periodic heat input can sustain oscillations (e.g., a Stirling engine). The first law links the supplied heat $Q_{\text{in}}$ to the work output $W_{\text{out}}$ and the change in internal energy:

$$Q_{\text{in}} = W_{\text{out}} + \Delta U$$

If the engine operates in a steady cyclic state, $\Delta U = 0$ and the efficiency is

$$\eta = \frac{W_{\text{out}}}{Q_{\text{in}}}$$

Common Misconceptions

• Confusing heat ($Q$) with temperature change. Heat is energy transfer; temperature is a state variable.
• Assuming that work done by the system is always positive. In damping, the system does negative work on its surroundings.
• Neglecting internal energy changes in processes where the temperature of the oscillator changes (e.g., heating a metal spring).

Sample A‑Level Question

Question: A mass $m = 0.5\;\text{kg}$ attached to a spring of constant $k = 200\;\text{N m}^{-1}$ oscillates with amplitude $A = 0.10\;\text{m}$ in air. The damping force is $F_{\text{damp}} = -0.05 v$. Determine the rate at which mechanical energy is converted to heat after one quarter of a period.

Solution Outline:

1. Calculate angular frequency: $\omega = \sqrt{k/m}$.
2. Find velocity at $t = T/4$: $v = \omega A$ (since $\sin(\omega t)=1$).
3. Power dissipated by damping: $P = F_{\text{damp}} v = -0.05 v^2$.
4. Insert the numerical values to obtain $P$ (the magnitude gives the rate of heat production).

Suggested Diagram