Short Notes: Oscillations
13.1 Simple Harmonic Motion
Simple harmonic motion (SHM) is a type of periodic motion in which the acceleration of the particle is always directed towards a fixed point (equilibrium) and is proportional to its displacement from that point. The equation of motion for a mass-spring system is
\(m\frac{d^{2}x}{dt^{2}} + kx = 0\),
whose general solution is
\(x = A\sin(\omega t + \phi)\) or \(x = A\cos(\omega t + \phi)\),
where \(A\) is the amplitude, \(\omega\) is the angular frequency and \(\phi\) is the phase constant. The angular frequency is related to the system parameters by
\(\omega = \sqrt{\dfrac{k}{m}}\) and \(\omega = 2\pi f = \dfrac{2\pi}{T}\).
| Parameter | Formula / Relationship |
|---|---|
| Displacement | \(x = A\sin(\omega t + \phi)\) or \(x = A\cos(\omega t + \phi)\) |
| Velocity | \(v = \dfrac{dx}{dt} = A\omega\cos(\omega t + \phi) = \pm \omega\sqrt{A^{2}-x^{2}}\) |
| Acceleration | \(a = \dfrac{d^{2}x}{dt^{2}} = -\omega^{2}x = -A\omega^{2}\sin(\omega t + \phi)\) |
| Time period | \(T = \dfrac{2\pi}{\omega}\) |
Phase relationships: Velocity is \(\dfrac{\pi}{2}\) (90°) out of phase with displacement and leads displacement by \(\dfrac{\pi}{2}\). Acceleration is in antiphase with displacement (180° out of phase) since \(a = -\omega^{2}x\).
Small-angle approximation for pendulum: For a simple pendulum of length \(L\), for small angular displacements (\(\theta \ll 1\) rad), \(\sin\theta \approx \theta\). This linearises the restoring torque and leads to SHM.
13.2 Energy in SHM
In SHM the mechanical energy oscillates between kinetic and potential forms while the total energy remains constant (in absence of damping). For a mass-spring system with spring constant \(k\) and mass \(m\):
| Energy type | Expression |
|---|---|
| Kinetic energy (KE) | \(\mathrm{KE} = \tfrac{1}{2}mv^{2} = \tfrac{1}{2}m\omega^{2}(A^{2} - x^{2})\) |
| Potential energy (PE) | \(\mathrm{PE} = \tfrac{1}{2}kx^{2} = \tfrac{1}{2}m\omega^{2}x^{2}\) |
| Total energy (constant) | \(E = \tfrac{1}{2}m\omega^{2}A^{2} = \tfrac{1}{2}kA^{2}\) |
At the mean position (\(x=0\)) the energy is purely kinetic and equals \(\tfrac{1}{2}m\omega^{2}A^{2}\). At the extreme positions (\(x=\pm A\)) the energy is purely potential and equals \(\tfrac{1}{2}kA^{2}\).
13.3 Time Periods of Different Systems
Common formulae for time period \(T\) of small oscillations for frequently encountered systems:
| System | Time period \(T\) |
|---|---|
| Horizontal spring-mass | \(T = 2\pi\sqrt{\dfrac{m}{k}}\) |
| Vertical spring-mass | \(T = 2\pi\sqrt{\dfrac{m}{k}} = 2\pi\sqrt{\dfrac{x_{0}}{g}}\) where \(x_{0}= \dfrac{mg}{k}\) is the static extension under the mass. |
| Simple pendulum (small angle) | \(T = 2\pi\sqrt{\dfrac{L}{g}}\) |
| Physical pendulum | \(T = 2\pi\sqrt{\dfrac{I}{mgd}}\) where \(I\) is moment of inertia about the pivot and \(d\) is distance from pivot to centre of mass. |
| Torsional pendulum | \(T = 2\pi\sqrt{\dfrac{I}{C}}\) where \(C\) is the torsional constant (torque per unit angle) and \(I\) is moment of inertia about the axis of twist. |
| Loaded spring (massive spring) | \(T = 2\pi\sqrt{\dfrac{m + m_{0}/3}{k}}\) where \(m_{0}\) is the mass of the spring (approximate correction for uniform spring). |
Notes: For the simple pendulum formula the result is independent of mass and valid only for small amplitudes. For a physical pendulum, a rigid body oscillates about a pivot and the effective restoring torque determines the period.
13.4 Damped Oscillations
When resistive forces (like friction or air resistance) act, the amplitude of oscillation decays with time. The linear damped oscillator equation is
\(m\frac{d^{2}x}{dt^{2}} + c\frac{dx}{dt} + kx = 0\),
where \(c\) is the damping constant. It is common to define the damping parameter \(b = \dfrac{c}{2m}\) and the natural (undamped) angular frequency \(\omega_{0} = \sqrt{\dfrac{k}{m}}\). The solution for the underdamped case is
\(x = A_{0}e^{-bt}\cos(\omega' t + \phi)\),
with
\(\omega' = \sqrt{\omega_{0}^{2} - b^{2}}\).
| Parameter / regime | Characteristic |
|---|---|
| Displacement (underdamped) | \(x = A_{0}e^{-bt}\cos(\omega' t + \phi)\) |
| Effective angular frequency | \(\omega' = \sqrt{\omega_{0}^{2} - b^{2}}\) |
| Types of damping | Underdamped (\(b<\omega_{0}\)): oscillatory="" decay.="" critically="" damped="" (\(b="\omega_{0}\)):" fastest="" return="" to="" equilibrium="" without="" oscillation.="" overdamped="" (\(b="">\omega_{0}\)): non-oscillatory slow return. |
Logarithmic decrement measures the rate of amplitude decay and is defined for successive amplitudes separated by one period. The quality factor \(Q\) is related to damping by \(Q = \dfrac{\omega_{0}}{2b}\); larger Q means weaker damping and sharper resonance.
13.5 Forced Oscillations and Resonance
- Forced oscillation: The system is driven by an external periodic force, typically \(F(t)=F_{0}\cos(\omega t)\).
- Steady-state amplitude response: For a driven damped oscillator the amplitude as a function of driving angular frequency \(\omega\) is
\(A(\omega) = \dfrac{F_{0}/m}{\sqrt{(\omega_{0}^{2}-\omega^{2})^{2} + (2b\omega)^{2}}}\). - Resonance: The amplitude is maximum near the natural frequency. For small damping the maximum occurs approximately at \(\omega \approx \omega_{0}\). For larger damping the resonance peak shifts to a slightly lower frequency; the exact resonance frequency is \(\omega_{r} = \sqrt{\omega_{0}^{2} - 2b^{2}}\) for the driven damped oscillator.
- Sharpness of resonance: Determined by damping. Less damping gives a higher peak and narrower width (larger Q), more damping gives a lower and broader peak (smaller Q).
- Practical implications: Resonance can be exploited (e.g., in musical instruments, radio tuners) and must be avoided in structures and machinery to prevent large amplitude destructive oscillations.
Examples and applications (brief): The simple pendulum illustrates SHM for small angles; clocks use pendulums or balance wheels (torsional oscillators) for timekeeping; damping is important in car suspensions and building design; resonance is crucial in designing bridges, buildings and electrical circuits (LC resonance).
Final summary: Simple harmonic motion is characterised by a sinusoidal displacement with constant frequency determined by system parameters. Energy oscillates between kinetic and potential with constant total energy in the absence of damping. Damping reduces amplitude over time and modifies frequency; forced driving can produce resonance where amplitude becomes large near a characteristic frequency. Understanding these relations, their formulae and phase relations is essential for analysing oscillatory systems.
FAQs on Short Notes: Oscillations
| 1. What are oscillations in physics? |  |
| 2. What is simple harmonic motion (SHM)? | |
| 3. How is the frequency of oscillation defined? | |
| 4. What is the significance of amplitude in oscillations? | |
| 5. Can you explain the concept of damping in oscillations? | |
