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Energy and Resonance
Energy Conservation in SHM
When a mass $m$ is displaced by $x_o$ against the restoring force of a spring, work is done that stores as elastic potential energy. This derivation uses the average force method.
$$W = \bar{F} \cdot d = \frac{1}{2}kx_o \times x_o = \frac{1}{2}kx_o^2$$
Work done against the spring equals the elastic potential energy stored.
$\bar{F}$=Average force during displacement(N)
$k$=Spring constant(N/m)
$x_o$=Maximum displacement (amplitude)(m)
Slow Stretching: The derivation assumes the spring is stretched slowly (zero acceleration), so the applied force equals the restoring force at every point.
Linear Force: Since Hooke's law gives $F = kx$, the force-displacement graph is a straight line — the work done is the triangular area under it.
Energy in Simple Harmonic Motion continuously oscillates between kinetic energy and elastic potential energy while the total mechanical energy remains constant.
$$E_{total} = \frac{1}{2}kA^2 = \text{constant}$$
The law of conservation of energy applied to an oscillating system.
$E_{total}$=Total mechanical energy(J)
$k$=Spring constant(N/m)
$A$=Amplitude of oscillation(m)
At equilibrium ($x=0$)
→Total energy is entirely kinetic: $E_{total} = \frac{1}{2}mv_{max}^2$
At amplitude ($x=\pm A$)
→Total energy is entirely potential: $E_{total} = \frac{1}{2}kA^2$
Equilibrium Position: Velocity is maximum and restoring force is zero, making K.E. maximum and P.E. zero.
Maximum Displacement: Velocity is zero and restoring force is maximum, making K.E. zero and P.E. maximum.
Pendulum Analogy: In a simple pendulum, gravitational P.E. at the top of the swing converts to K.E. at the bottom — same interchange, different energy type.
Proportionality Shortcut: Since $E \propto A^2$, if amplitude drops by half, energy drops to one-quarter.
At any displacement $x$, the energy splits into kinetic energy (motion) and elastic potential energy (deformation), and their sum always equals the total.
$$P.E. = \frac{1}{2}kx^2 \quad ; \quad K.E. = \frac{1}{2}k(A^2 - x^2)$$
The position-dependent energy partition in SHM.
$x$=Instantaneous displacement from equilibrium(m)
$A$=Amplitude(m)
$k$=Spring constant(N/m)
$x = A/2$
→P.E. = $\frac{1}{4}E_{total}$, K.E. = $\frac{3}{4}E_{total}$
$x = A/\sqrt{2}$
→P.E. = K.E. = $\frac{1}{2}E_{total}$ (the equal-split point)
Ratio Shortcut: To find P.E. at any position, just compute $(x/A)^2$ and multiply by $E_{total}$. No need to know $k$ or $m$ separately.
Velocity at Position x: From K.E. = $\frac{1}{2}mv^2$, the velocity is $v = v_{max}\sqrt{1 - x^2/A^2}$.
Dimensional Check: Both $\frac{1}{2}kx^2$ and $\frac{1}{2}mv^2$ have dimensions $[M][L]^2[T]^{-2}$ = Joules.
Energy-Displacement Graphs
The relationship between energy and displacement follows a parabolic relationship, with the total energy represented by a horizontal line.
$$K.E. = \frac{1}{2}k(A^2 - x^2)$$
Kinetic energy as a function of position — an inverted parabola.
$k$=Spring constant(N/m)
$A$=Amplitude(m)
$x$=Instantaneous displacement(m)
$x = 0$ (equilibrium)
→K.E. = $\frac{1}{2}kA^2$ = maximum
$x = \pm A$ (extremes)
→K.E. = 0, object momentarily at rest
P.E. Curve: Upward-opening parabola ($\frac{1}{2}kx^2$) — minimum at center, maximum at edges.
K.E. Curve: Downward-opening parabola — maximum at center, zero at edges.
Equal Energy Point: At $x = \pm A/\sqrt{2} \approx \pm 0.707A$, the K.E. and P.E. curves cross. This is the only position where energy is split equally.
Total Energy Line: A flat horizontal line at $E = \frac{1}{2}kA^2$, confirming conservation.
Free and Forced Oscillations
A body executing free oscillations vibrates without interference from an external force, oscillating at its natural frequency determined solely by the system's physical properties.
Natural Frequency: For a spring-mass system, $f_0 = \frac{1}{2\pi}\sqrt{k/m}$. For a pendulum, $f_0 = \frac{1}{2\pi}\sqrt{g/l}$.
Independence from Amplitude: The natural frequency of ideal SHM does not depend on amplitude — only on system parameters ($k$, $m$, or $l$, $g$).
Real Systems: Free oscillations gradually die out due to friction and air resistance dissipating energy.
When a freely oscillating system is subjected to an external periodic force, forced oscillations take place. The system is then called a driven harmonic oscillator.
Driving Force: An external periodic force that continuously supplies energy to the system.
Steady State: After initial transients die out, the system oscillates at the driving frequency, not its natural frequency.
Examples: Vehicle vibrations from engine, loud music from wooden boards of string instruments, pushing a swing periodically.
Resonance
Resonance occurs when the driving frequency of an external periodic force equals the natural frequency of the oscillator, causing the amplitude to become extraordinarily large.
$$f_{driving} = f_0$$
The condition for maximum energy transfer and maximum amplitude.
$f_{driving}$=Frequency of the external periodic force(Hz)
$f_0$=Natural frequency of the system(Hz)
$f_{driving} \ll f_0$
→Amplitude is small — the system barely responds to the slow driving
$f_{driving} = f_0$
→Maximum amplitude — energy transfer from driving source is most efficient
$f_{driving} \gg f_0$
→Amplitude is small — the system cannot keep up with the fast driving
Maximum Energy Transfer: At resonance, the driving force is perfectly synchronized with the oscillation — each push adds energy at the optimal moment.
Phase at Resonance: The velocity of the oscillator is in phase with the driving force, maximizing power transfer.
Coupled Pendulum Demo: Pendulums of matching length on a shared rod transfer energy efficiently; non-matching ones remain small — a classic demonstration of resonance selectivity.
Resonance has both beneficial applications and dangerous consequences in real-world systems.
Radio Tuning: Turning the dial changes the circuit's natural frequency to match the broadcast frequency — at resonance, energy absorption from the signal is maximum and only that station is heard.
Microwave Oven: Produces waves at 2450 MHz (wavelength 12 cm) that match the resonant frequency of water and fat molecules, causing them to absorb energy and heat the food.
Soldiers on Bridge: Marching soldiers break step on long bridges because rhythmic marching could match the bridge's natural frequency and build up dangerously large oscillations.
Tacoma Narrows Bridge: Collapsed due to wind-driven oscillations matching the bridge's natural frequency — a dramatic example of destructive resonance.
Damped Oscillations
Damped oscillations are oscillations whose amplitude decreases steadily with time because resistive forces (friction, air resistance, viscous drag) dissipate energy from the system.
$$E \propto A^2$$
Since energy is proportional to amplitude squared, any reduction in amplitude means energy has been lost to resistive forces.
$E$=Remaining mechanical energy(J)
$A$=Current amplitude (decreasing over time)(m)
Ideal vs Real: Pure SHM (constant amplitude forever) is an idealization. All real oscillators experience damping.
Light Damping: System oscillates many times with a slowly decaying exponential decay amplitude envelope.
Critical Damping: System returns to equilibrium in the shortest possible time without overshooting — used in car shock absorbers to quickly suppress oscillations after a bump.
Overdamping: Very high resistance causes an extremely slow, non-oscillatory return to equilibrium.
Limiting Case: As damping → 0, the system approaches ideal SHM with constant amplitude.
Sharpness of Resonance
The sharpness of resonance describes how rapidly the amplitude drops when the driving frequency deviates from the natural frequency. It depends inversely on damping.
Sharp Resonance (low damping): Tall, narrow peak — amplitude is very large at $f_0$ but drops off rapidly. A lead-bob pendulum shows this behavior due to low air resistance relative to its weight.
Flat Resonance (high damping): Short, wide peak — moderate amplitude over a broad frequency range. A pith-ball pendulum shows this due to high air resistance relative to its light weight.
Practical Implication: In radio circuits, sharp resonance allows tuning to a single station without interference from neighboring frequencies.