Simple Harmonic Motion (SHM) is a core part of Topic 17 in the AS‑level syllabus and underpins many A‑level extensions (resonance, quality factor, coupled oscillators, phase‑space analysis). Mastery of the definitions, mathematics and experimental techniques is essential for AO1, AO2 and AO3 assessment objectives.
\[
T=\frac{2\pi}{\omega}=\frac{1}{f}
\]
\[
f=\frac{1}{T}=\frac{\omega}{2\pi}
\]
\[
\omega = 2\pi f = \frac{2\pi}{T}
\]
\[
Q=\frac{\omega_0}{2\gamma},
\]
where \(\omega_0\) is the natural angular frequency and \(\gamma\) the damping constant.
\[
m\ddot{x}=-kx\;\Longrightarrow\;\ddot{x}+\frac{k}{m}x=0.
\]
\[
\boxed{\;\omega=\sqrt{\dfrac{k}{m}}\;}
\]
giving the standard SHM differential equation
\[
\ddot{x}=-\omega^{2}x.
\]
Two equivalent forms are used in the exam:
\[
x(t)=A\cos(\omega t+\phi)\qquad\text{or}\qquad x(t)=A\sin(\omega t+\phi).
\]
From the initial displacement \(x0=x(0)\) and initial velocity \(v0=\dot{x}(0)\):
\[
\begin{aligned}
x_0 &= A\cos\phi,\\
v_0 &= -A\omega\sin\phi.
\end{aligned}
\]
Hence
\[
\phi=\tan^{-1}\!\left(-\frac{v0}{\omega x0}\right),\qquad
A=\sqrt{x0^{2}+\left(\frac{v0}{\omega}\right)^{2}}.
\]
\[
x(t)=\Re\!\left[\,Ce^{i\omega t}\,\right],\qquad C=Ae^{i\phi}.
\]
This compact notation is handy when adding a driving force or damping term.
\[
\omega T = 2\pi \;\Longrightarrow\; T=\frac{2\pi}{\omega}.
\]
\[
f=\frac{\omega}{2\pi}.
\]
\[
T=\frac{1}{f},\qquad \omega=2\pi f,\qquad T=\frac{2\pi}{\omega}.
\]
| Quantity | Expression | Comment |
|---|---|---|
| Kinetic energy, \(K\) | \(\displaystyle K=\frac12 m\dot{x}^{2}= \frac12 m\omega^{2}A^{2}\sin^{2}(\omega t+\phi)\) | Maximum \(K_{\max}= \frac12 m\omega^{2}A^{2}\) at equilibrium. |
| Potential energy, \(U\) | \(\displaystyle U=\frac12 kx^{2}= \frac12 m\omega^{2}A^{2}\cos^{2}(\omega t+\phi)\) | Maximum \(U_{\max}= \frac12 m\omega^{2}A^{2}\) at extreme positions. |
| Total mechanical energy, \(E\) | \(\displaystyle E=K+U=\frac12 m\omega^{2}A^{2}\) (constant) | Independent of time; useful for AO2 energy‑conservation questions. |
For a damping force \(F_{d}=-b\dot{x}\) the mechanical energy decays exponentially:
\[
E(t)=E_0\,e^{-2\gamma t},\qquad \gamma=\frac{b}{2m}.
\]
The lost energy appears as heat, a point often required in AO2 explanations.
The equation of motion becomes
\[
\ddot{x}+2\gamma\dot{x}+\omega_0^{2}x=0,
\]
where \(\omega_0=\sqrt{k/m}\) is the natural angular frequency.
\[
x(t)=Ae^{-\gamma t}\cos(\omega' t+\phi),\qquad
\omega'=\sqrt{\omega_0^{2}-\gamma^{2}}.
\]
\[
\Delta\omega=\frac{\omega_0}{Q}.
\]
When an external periodic force \(F{\text{d}}=F0\cos(\omega_{\text{d}}t)\) acts, the steady‑state solution is
\[
x(t)=A{\text{d}}\cos(\omega{\text{d}}t-\delta),
\]
with
\[
A{\text{d}}=\frac{F0/m}{\sqrt{(\omega0^{2}-\omega{\text{d}}^{2})^{2}+ (2\gamma\omega_{\text{d}})^{2}}},\qquad
\tan\delta=\frac{2\gamma\omega{\text{d}}}{\omega0^{2}-\omega_{\text{d}}^{2}}.
\]
For an undamped oscillator the trajectory in the \((x,v)\) plane is an ellipse:
\[
\frac{x^{2}}{A^{2}}+\frac{v^{2}}{(\omega A)^{2}}=1.
\]
When damping is present the ellipse spirals inward, illustrating the loss of mechanical energy.
Problem: A 0.50 kg mass is attached to a spring with force constant \(k=200\;\text{N m}^{-1}\). The mass is pulled 0.04 m to the right and released from rest.
Solution:
\[
\omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{200}{0.50}}=20\;\text{rad s}^{-1},
\qquad
T=\frac{2\pi}{\omega}= \frac{2\pi}{20}=0.314\;\text{s}.
\]
Maximum speed occurs at equilibrium:
\[
v_{\max}=A\omega=0.04\times20=0.80\;\text{m s}^{-1}.
\]
Maximum kinetic energy:
\[
K{\max}=\frac12 m v{\max}^{2}= \frac12(0.50)(0.80)^{2}=0.16\;\text{J}.
\]
Total mechanical energy (constant):
\[
E=\frac12 m\omega^{2}A^{2}= \frac12(0.50)(20)^{2}(0.04)^{2}=0.16\;\text{J}.
\]
(Notice \(E=K{\max}=U{\max}\).)
For \(x(t)=A\cos(\omega t+\phi)\) with the mass released from rest at \(x=A\), we have \(\phi=0\). Setting \(x=0\):
\[
0=A\cos(\omega t)\;\Longrightarrow\;\omega t=\frac{\pi}{2}.
\]
Hence
\[
t_{\text{eq}}=\frac{\pi}{2\omega}= \frac{\pi}{40}=0.0785\;\text{s}.
\]
Because the initial conditions are \(x(0)=A\) and \(\dot{x}(0)=0\),
\[
\phi=0\quad\text{(or }2\pi\text{)}.
\]
| Quantity | Symbol | SI Unit | Definition / Relationship |
|---|---|---|---|
| Displacement | \(x\) | m | Instantaneous distance from equilibrium |
| Amplitude | \(A\) | m | Maximum \(|x|\) |
| Period | \(T\) | s | \(T=2\pi/\omega=1/f\) |
| Frequency | \(f\) | Hz | \(f=1/T=\omega/2\pi\) |
| Angular frequency | \(\omega\) | rad s\(^{-1}\) | \(\omega=2\pi f=2\pi/T\); for a spring \(\omega=\sqrt{k/m}\) |
| Phase constant | \(\phi\) | rad | Sets the initial condition; \(\phi=\tan^{-1}\!\bigl(-v0/(\omega x0)\bigr)\) |
| Phase difference | \(\Delta\phi\) | rad | Angular offset between two SHM motions of the same \(\omega\) |
| Quality factor | \(Q\) | – | \(Q=\omega_0/(2\gamma)\); higher \(Q\) ⇒ sharper resonance |
| Total mechanical energy | \(E\) | J | \(E=\tfrac12 m\omega^{2}A^{2}\) (constant for undamped SHM) |
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