analyse and interpret graphical representations of the variations of displacement, velocity and acceleration for simple harmonic motion

Simple Harmonic Motion (Cambridge International AS & A Level Physics 9702 – Sections 17.1‑17.3)

Learning Objective (AO1 + AO2)

Analyse and interpret the graphical representations of displacement, velocity and acceleration for simple harmonic motion (SHM), and extend the analysis to energy, damping and resonance.

1. Core Concepts of Undamped SHM

• Definition (AO1): Periodic motion in which the restoring force is directly proportional to the displacement from equilibrium and always directed towards it:

  \[

  F_{\text{restore}}=-kx \qquad\Longrightarrow\qquad a=-\frac{k}{m}x .

• Key relation (AO1): From the above, the angular frequency is

  \[

  \boxed{\;\omega=\sqrt{\frac{k}{m}}\;},\qquad\text{or}\qquad k=m\omega^{2}.

• Mathematical description (AO1):

  • Displacement \(x(t)=A\cos(\omega t+\phi)\)  \(A\,[\text{m}],\;\omega\,[\text{rad s}^{-1}],\;\phi\,[\text{rad}]\)
  • Velocity \(v(t)=\frac{dx}{dt}=-A\omega\sin(\omega t+\phi)\)
  • Acceleration \(a(t)=\frac{dv}{dt}=-A\omega^{2}\cos(\omega t+\phi)=-\omega^{2}x(t)\)

• Derivation by differentiation (AO2): Shows the inter‑relationship \(a=-\omega^{2}x\) and that velocity is the time‑derivative of displacement.
• Integration note (AO2): If acceleration is given, integrate once to obtain \(v(t)\) (adding constant \(C{1}\)), then integrate again to obtain \(x(t)\) (adding constant \(C{2}\)). The two constants combine to give the phase constant \(\phi\).

1.1 Parameters and How to Obtain Them from a Graph (AO2)

1.2 Graphical Relationships (AO2)

• Displacement vs. time – cosine (or sine) wave of amplitude \(A\) and period \(T\).
• Velocity vs. time – sine wave, shifted by \(\frac{T}{4}\) (or \(90^{\circ}\)) relative to displacement; amplitude \(A\omega\).
• Acceleration vs. time – cosine wave identical in shape to displacement but inverted; amplitude \(A\omega^{2}\).

All three graphs share the same period \(T\); the phase relationships are summarised below.

1.3 Extracting \(\omega\) from a Graph (AO2)

1. Measure the period \(T\) from any repeating feature.
2. Calculate \(\displaystyle\omega=\frac{2\pi}{T}\).
3. Check consistency:

   • Peak‑to‑peak vertical distance on the velocity graph should be \(2A\omega\).
   • Peak‑to‑peak vertical distance on the acceleration graph should be \(2A\omega^{2}\).

1.4 Worked Example (Undamped) – AO2

A mass–spring system oscillates with amplitude \(A=0.05\;\text{m}\) and angular frequency \(\omega=10\;\text{rad s}^{-1}\). Take \(\phi=0\) for simplicity.

1. Displacement: \(x(t)=0.05\cos(10t)\) m
2. Velocity: \(v(t)=-0.05\times10\sin(10t)=-0.5\sin(10t)\) m s⁻¹
3. Acceleration: \(a(t)=-0.05\times10^{2}\cos(10t)=-5\cos(10t)\) m s⁻²
4. Period: \(T=\dfrac{2\pi}{\omega}=0.628\;\text{s}\)
5. Maximum speed: \(v_{\max}=A\omega=0.5\;\text{m s}^{-1}\)
6. Maximum acceleration: \(a_{\max}=A\omega^{2}=5\;\text{m s}^{-2}\)
7. Energy check (using \(k=m\omega^{2}\)):

   • Assume \(m=0.20\;\text{kg}\) ⇒ \(k=m\omega^{2}=0.20\times100=20\;\text{N m}^{-1}\)
   • Total mechanical energy \(E=\frac12kA^{2}= \frac12(20)(0.05)^{2}=0.025\;\text{J}\)
   • At any instant \(E=\frac12mv^{2}+\frac12kx^{2}\) – a useful AO2 verification.

2. Energy in Simple Harmonic Motion (AO1 + AO2)

• Kinetic energy: \(K=\frac12mv^{2}\)
• Elastic potential energy: \(U=\frac12kx^{2}\) (with \(k=m\omega^{2}\))
• Total mechanical energy (conserved for undamped SHM):

  \[

  E = K{\max}=U{\max}= \frac12kA^{2}= \frac12m\omega^{2}A^{2}

  \]

• Energy oscillates between kinetic and potential with the same period \(T\); a plot of \(E\) versus \(t\) is a sinusoid of double frequency (peaks when \(x=0\) or \(v=0\)).

3. Damped and Forced Oscillations (Section 17.3)

3.1 Damping Regimes (AO1 + AO2)

Definitions:

• \(\beta\) – damping coefficient (s⁻¹).
• \(\omega_{0}=\sqrt{k/m}\) – natural angular frequency of the undamped system.
• Observed angular frequency for the under‑damped case: \(\omega'=\sqrt{\omega_{0}^{2}-\beta^{2}}\); the corresponding period is \(T'=\dfrac{2\pi}{\omega'}\) (slightly longer than the undamped period).

3.2 Forced Oscillations & Resonance (AO1 + AO2)

• Driving force: \(F{\text{drive}} = F{0}\cos(\omega{\!d}t)\) where \(\omega{\!d}\) is the driving angular frequency.
• The steady‑state solution has the same frequency as the driver:

  \[

  x(t)=X(\omega{\!d})\cos(\omega{\!d}t-\delta)

  \]

  with

  \[

  X(\omega{\!d})=\frac{F{0}/m}{\sqrt{(\omega{0}^{2}-\omega{\!d}^{2})^{2}+ (2\beta\omega_{\!d})^{2}}},\qquad

  \tan\delta=\frac{2\beta\omega{\!d}}{\omega{0}^{2}-\omega_{\!d}^{2}} .

  \]

• Resonance: For light damping the amplitude is maximum when \(\omega{\!d}\approx\omega{0}\). The resonance curve (amplitude vs. \(\omega_{\!d}\)) is a key AO1/AO2 topic.
• At resonance the energy supplied per cycle equals the energy lost to damping, so amplitudes can become very large if the system is not heavily damped.

4. Systematic Procedure for Interpreting Real Graphs (AO2)

1. Identify the period \(T\) from any repeating feature (peak‑to‑peak, zero‑crossing, etc.).
2. Determine the amplitude \(A\) from the displacement graph.
3. Calculate \(\omega\) using \(\omega=2\pi/T\). Use the key relation \(k=m\omega^{2}\) if the spring constant is required.
4. Find the phase constant \(\phi\) by comparing the first peak (or zero‑crossing) with a reference cosine: \(\phi=\omega\Delta t\).
5. Assess damping: If successive peaks decrease, fit an exponential envelope \(A\,e^{-\beta t}\) to estimate \(\beta\). Compare the observed period with the undamped period to see the slight lengthening.
6. Energy verification: Compute \(K\) and \(U\) at several points; the sum should be constant for undamped data and slowly decreasing for damped data.

5. Common Misconceptions (and How to Address Them) (AO1 + AO2)

• Amplitude vs. maximum speed: Remember \(v_{\max}=A\omega\); the angular frequency factor is often omitted.
• Phase difference: Velocity leads displacement by \(90^{\circ}\) (or lags by \(-90^{\circ}\)); acceleration is \(180^{\circ}\) out of phase with displacement.
• Direction of acceleration: Acceleration always points towards the equilibrium position, not opposite the instantaneous velocity.
• Effect of damping on period: Light damping lengthens the period slightly; the period is *not* strictly unchanged.
• Resonance condition: Resonance occurs at the natural frequency of the *undamped* system, not at the driving frequency that gives the largest observed amplitude when heavy damping is present.

6. Summary Checklist (AO1 + AO2)

• Write the three SHM equations and identify \(A\), \(\omega\), \(\phi\), and \(T\) (AO1).
• Use the key relation \(k=m\omega^{2}\) to link spring constant and angular frequency (AO1).
• Recall amplitude relationships: \(v{\max}=A\omega\), \(a{\max}=A\omega^{2}\) (AO2).
• State phase relationships: \(v\) is \(90^{\circ}\) out of phase, \(a\) is \(180^{\circ}\) out of phase with \(x\) (AO2).
• Apply the energy formula \(E=\tfrac12m\omega^{2}A^{2}\) to check consistency (AO2).
• Identify the three damping regimes and write the appropriate displacement expression (AO1).
• Understand the forced‑oscillation amplitude formula and the condition for resonance (AO1 + AO2).
• Follow the systematic graph‑analysis routine: period → \(\omega\) → amplitudes → phase constant → damping (AO2).

7. Practice Questions (AO1 + AO2)

1. Undamped SHM
   

   A displacement graph shows a peak‑to‑peak amplitude of \(0.40\;\text{m}\) and a period of \(2.0\;\text{s}\).

   • (a) Determine \(A\), \(\omega\), \(v{\max}\) and \(a{\max}\).
   • (b) Sketch the corresponding velocity and acceleration graphs, indicating zero‑crossings and extrema.

2. Phase constant from a graph
   

   The first maximum of the displacement curve occurs at \(t=0.35\;\text{s}\). The measured period is \(T=1.20\;\text{s}\). With \(A=0.06\;\text{m}\), find \(\phi\) (rad and degrees) and write the explicit expression for \(x(t)\).

3. Energy verification
   

   For the system in Question 1, the mass is \(0.25\;\text{kg}\). Calculate the total mechanical energy and show that at \(t=0.25\;\text{s}\) the sum of kinetic and potential energies equals this value.

4. Damped oscillation
   

   An under‑damped oscillator follows \(x(t)=0.08\,e^{-0.5t}\cos(6t)\) (SI units).

   • (a) Write the expressions for \(v(t)\) and \(a(t)\).
   • (b) Determine \(\beta\), the natural frequency \(\omega_{0}\), and the observed angular frequency \(\omega'\).
   • (c) Sketch the exponential envelope \(\pm0.08e^{-0.5t}\) on a graph of \(x\) versus \(t\).

5. Forced resonance
   

   A driver exerts a force \(F(t)=2\cos(\omega_{\!d}t)\) N on a mass‑spring system with \(m=0.10\;\text{kg}\) and \(k=20\;\text{N m}^{-1}\). The damping coefficient is \(\beta=0.2\;\text{s}^{-1}\).

   • (a) Calculate the natural frequency \(\omega_{0}\).
   • (b) Find the driving frequency \(\omega_{\!d}\) that gives the maximum steady‑state amplitude (to two significant figures).
   • (c) State the corresponding phase lag \(\delta\) (in degrees).