16 Thermodynamics – Internal Energy and Work
16.1 Internal Energy (U)
- Definition: the total microscopic kinetic + potential energy of all particles in a system.
- State function: U depends only on the current state (T, p, V, n…) and not on the path taken to reach that state.
- Kinetic‑theory view (ideal gas): for a monatomic ideal gas the microscopic kinetic energy of the molecules gives
$$U = \tfrac{3}{2}\,nRT$$
where n is the amount of gas, R the universal gas constant and T the absolute temperature.
- Temperature dependence: for any ideal gas
$$U = nC_{V}T$$
where CV is the molar heat capacity at constant volume.
Hence a change in temperature directly changes the internal energy:
$$\Delta U = nC_{V}\,\Delta T$$
- Heat‑capacity relationship (Cambridge 9702):
$$C_{P}=C_{V}+R$$
(useful when a process occurs at constant pressure).
- Independence of pressure/volume (ideal gas): because U is a function of T only, changing p or V at constant T does not change U.
16.2 Work Done at Constant Pressure
When a gas expands or contracts against a constant external pressure p, the work done on the system is
$$W = -p\Delta V$$
- Cambridge sign convention: W is **positive** when work is done **on** the system (compression) and **negative** when the system does work on the surroundings (expansion).
- First‑law form used in the syllabus:
$$\Delta U = q + W$$
where q is the heat added to the system (positive when heat flows into the system).
First Law of Thermodynamics (syllabus form)
$$\boxed{\Delta U = q + W}$$
- Isochoric process (ΔV = 0): \(W=0\) ⇒ \(\Delta U = q\).
- Isobaric process (constant p): substitute \(W = -p\Delta V\) into the first‑law equation.
- Isothermal ideal‑gas expansion: \(\Delta U = 0\) (since \(T\) constant) ⇒ \(q = -W = p\Delta V\).
17 Oscillations – Simple Harmonic Motion (SHM) and Damping
17.1 Simple Harmonic Oscillator (SHO) – Core Equations
| Quantity | Symbol | Expression |
|---|
| Displacement from equilibrium |
\(x\) |
\(x = x_{0}\sin(\omega t + \phi)\) |
| Amplitude |
\(x_{0}\) |
Maximum displacement (positive constant) |
| Angular frequency |
\(\omega\) |
\(\displaystyle\omega = 2\pi f = \sqrt{\frac{k}{m}}\) |
| Period |
\(T\) |
\(\displaystyle T = \frac{2\pi}{\omega}\) |
| Velocity |
\(v\) |
\(\displaystyle v = \frac{dx}{dt}= \omega x_{0}\cos(\omega t + \phi)\) |
| Acceleration |
\(a\) |
\(\displaystyle a = \frac{d^{2}x}{dt^{2}} = -\omega^{2}x\) |
| Phase constant |
\(\phi\) |
Sets the starting point of the motion (determined from initial conditions) |
Energy in an Undamped SHO
- Kinetic energy
$$K = \tfrac12 mv^{2}= \tfrac12 m\omega^{2}\bigl(x_{0}^{2}-x^{2}\bigr)$$
- Elastic potential energy of the spring
$$U_{s}= \tfrac12 kx^{2}= \tfrac12 m\omega^{2}x^{2}$$
- Total mechanical energy (constant)
$$E_{\text{mech}} = K+U_{s}= \tfrac12 m\omega^{2}x_{0}^{2}= \tfrac12 kx_{0}^{2}$$
17.2 Damped Oscillations
A real oscillator experiences a resistive force that opposes the motion. In the Cambridge syllabus the damping force is taken as proportional to the velocity:
$$F_{\text{damp}} = -b\,v$$
- b – damping constant (units kg s⁻¹). The negative sign ensures the force always acts opposite to the direction of motion.
- The damping force does **negative work** on the oscillator, removing mechanical energy and converting it to heat in the surroundings.
Power and Energy Loss
The instantaneous power supplied by the damping force is
$$P = F_{\text{damp}}\,v = -b\,v^{2}$$
- The negative sign shows that mechanical energy is being removed at a rate \(|P|\).
- Applying the first law to the combined system (oscillator + surroundings):
$$\Delta U_{\text{sur}} = q = -W_{\text{damp}} = b\,v^{2}$$
where \(\Delta U_{\text{sur}}\) is the increase in internal energy (heat) of the surroundings.
Types of Damping (Cambridge 9702 17.3)
| Type | Characteristic displacement‑time graph |
|---|
| Light (underdamped) |
Oscillatory motion with an exponentially decaying envelope; the peaks follow \(x_{\max}=x_{0}e^{-bt/2m}\). |
| Critical (critically damped) |
Returns to equilibrium as quickly as possible without overshooting; the graph is a single exponential decay, \(x(t)=A\,t\,e^{-\omega_{0}t}\) (no sinusoidal term). |
| Heavy (overdamped) |
Slow, monotonic return to equilibrium; the decay is slower than the critically damped case and no oscillation occurs. |
Mathematical form (optional for A‑Level)
For a linear damped SHO the displacement can be written as
$$x(t)=A\,e^{-\,\frac{b}{2m}t}\sin(\omega' t+\phi),\qquad
\omega'=\sqrt{\omega^{2}-\left(\frac{b}{2m}\right)^{2}}$$
where \(\omega' \) is the **damped angular frequency**. When \(b/2m \ll \omega\) the motion is essentially the same as the undamped case.
17.3 Resonance (Driven Oscillations)
- A periodic driving force \(F_{\text{drive}} = F_{0}\sin(\omega_{d}t)\) is applied to the oscillator.
- The steady‑state amplitude is
$$A(\omega_{d}) = \frac{F_{0}}{m\sqrt{(\omega^{2}-\omega_{d}^{2})^{2}+(\frac{b\omega_{d}}{m})^{2}}}$$
- Resonance condition: the amplitude reaches a maximum when the driving angular frequency \(\omega_{d}\) equals the natural (undamped) angular frequency \(\omega\) (or is very close to it for light damping).
- At resonance the phase difference between the driving force and the displacement is \(\pi/2\) rad (90°).
- In the syllabus this is identified by a peak in an amplitude‑versus‑frequency graph.
17.4 Models of Physical Systems
- Mass‑spring system: restoring force \(F=-kx\).
- Simple pendulum (small angles): restoring torque \(\tau\approx -mgL\,\theta\) ⇒ SHM with \(\omega=\sqrt{g/L}\).
- LC electrical circuit: charge‑displacement analogue; restoring “force” provided by the inductor, \(\omega=1/\sqrt{LC}\).
- Damping models:
- Air resistance (linear: \(-b v\)) – syllabus approved approximation.
- Viscous friction in fluids.
- Electrical resistance in an RLC circuit (analogue of \(-b v\)).
Energy Transfer Summary (Thermodynamics ↔ Oscillations)
| Process | Energy lost | Energy gained | Syllabus link |
|---|
| Damped SHO |
Mechanical (K + Us) |
Internal energy of surroundings (heat) |
16.1, 17.2 |
| Isobaric expansion of a gas |
Internal energy (if temperature falls) |
Work on surroundings + heat flow |
16.2 |
| Resonantly driven oscillator |
External work supplied (periodic force) |
Mechanical energy (maintained amplitude) |
17.3 |
Sample A‑Level Question (Cambridge style)
Question: A mass \(m = 0.5\;\text{kg}\) is attached to a spring of constant \(k = 200\;\text{N m}^{-1}\). The system oscillates in air with a damping force \(F_{\text{damp}} = -0.05\,v\). The amplitude at \(t = 0\) is \(A = 0.10\;\text{m}\). Calculate the rate at which mechanical energy is being converted to heat when the mass passes through the equilibrium position for the first time.
Solution (step‑by‑step, AO2 style)
- Angular frequency (ignoring damping for the speed at equilibrium):
\[
\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.5}} = 20\;\text{rad s}^{-1}
\]
- Maximum speed at the equilibrium position:
\[
v_{\max}= \omega A = 20 \times 0.10 = 2.0\;\text{m s}^{-1}
\]
- Power dissipated by the damping force:
\[
P = -F_{\text{damp}}\,v = -(-0.05\,v)\,v = 0.05\,v^{2}
\]
- Insert \(v = v_{\max}\):
\[
P = 0.05 \times (2.0)^{2} = 0.20\;\text{W}
\]
- Interpretation: 0.20 J of mechanical energy is converted to heat each second when the mass passes through the equilibrium point.
Practical Skills Linked to the Syllabus
- Measuring period and frequency: Use a stopwatch, photogate, or motion‑sensor software to record displacement‑time data and obtain \(T\) and \(f\).
- Determining the damping constant \(b\): Record successive amplitudes \(A_{1},A_{2},\dots\); plot \(\ln A\) against time. The gradient equals \(-b/(2m)\) (linear‑fit method).
- Heat measurement in a damped system: Place a small calorimeter or thermistor in the surrounding air; the temperature rise \(\Delta T\) gives the heat \(q = C_{\text{cal}}\Delta T\), confirming the energy‑loss calculation.
- Resonance experiment: Drive a mass‑spring system with a function generator; vary the driving frequency and plot amplitude versus frequency to locate the resonance peak.
Assessment Objectives Covered
- AO1 – Knowledge: Definitions of internal energy, heat capacities, SHM equations, damping types, resonance.
- AO2 – Application: Use \(\Delta U = q + W\), \(F_{\text{damp}}=-b v\), and energy‑conservation formulas to solve quantitative problems.
- AO3 – Analysis: Interpret displacement‑time graphs to distinguish light, critical and heavy damping; analyse amplitude‑frequency data for resonance.
Key Points to Remember
- Internal energy is the microscopic kinetic + potential energy; for an ideal gas it depends **only** on temperature (U ∝ T).
- Heat capacities: \(C_{P}=C_{V}+R\); use \(C_{V}\) for constant‑volume processes and \(C_{P}\) for constant‑pressure processes.
- Cambridge sign convention in the first law: \(\Delta U = q + W\) (both \(q\) and \(W\) positive when they add energy to the system).
- In a SHO without non‑conservative forces, total mechanical energy is constant: \(E_{\text{mech}}=\tfrac12 kx_{0}^{2}\).
- Damping removes mechanical energy at a rate \(P=-b v^{2}\); the lost energy appears as heat, raising the internal energy of the surroundings.
- Resonance occurs when the driving frequency equals the natural frequency; the amplitude peaks and the phase difference is \(\pi/2\) rad.