🌊 Introduction to Simple Harmonic Motion

What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion is periodic oscillatory motion where the restoring force is proportional to displacement:

$F = -kx$

where:

• $F$ = restoring force (N)
• $k$ = spring constant or force constant (N/m)
• $x$ = displacement from equilibrium (m)
• Negative sign: force always points toward equilibrium

💡 Key Idea: SHM occurs when an object experiences a restoring force proportional to its displacement from equilibrium. The negative sign means the force always tries to bring the object back to equilibrium.

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Examples of SHM

Mass-Spring System:

• Mass attached to spring
• Pull and release → oscillates
• Spring force: $F = -kx$ (Hooke's Law)

Simple Pendulum:

• Mass on string
• Pull aside and release → swings
• For small angles: acts like SHM
• Restoring force: component of gravity

Other Examples:

• Vibrating guitar string
• Atoms in a solid (vibrate about equilibrium)
• Molecular bonds
• Earthquake oscillations (seismograph)

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Hooke's Law

For an ideal spring, the force is proportional to displacement:

$F = -kx$

Spring Constant $k$:

• Measures spring "stiffness"
• Units: N/m
• Large $k$: stiff spring (hard to stretch)
• Small $k$: soft spring (easy to stretch)

Equilibrium Position:

• Where spring is unstretched ($x = 0$)
• No net force
• Object naturally returns here

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Characteristics of SHM

Amplitude $A$

Maximum displacement from equilibrium:

• If object goes from $-A$ to $+A$, amplitude is $A$
• Always positive
• Units: meters
• Independent of mass or spring constant

Period $T$

Time for one complete oscillation:

• From max displacement, through equilibrium, to other max, and back
• Units: seconds (s)
• Does NOT depend on amplitude!

For mass-spring system: $T = 2\pi\sqrt{\frac{m}{k}}$

For simple pendulum (small angles): $T = 2\pi\sqrt{\frac{L}{g}}$

where $L$ is length, $g$ is gravitational acceleration.

Frequency $f$

Number of oscillations per second: $f = \frac{1}{T}$

Units: Hz (hertz) = 1/s = cycles/second

For mass-spring: $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Angular Frequency $\omega$

$\omega = 2\pi f = \frac{2\pi}{T}$

Units: rad/s

For mass-spring: $\omega = \sqrt{\frac{k}{m}}$

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Position, Velocity, and Acceleration in SHM

Position

$x(t) = A\cos(\omega t + \phi)$

or

$x(t) = A\sin(\omega t + \phi)$

where:

• $A$ = amplitude
• $\omega$ = angular frequency
• $\phi$ = phase constant (depends on initial conditions)
• $t$ = time

Velocity

$v(t) = -A\omega\sin(\omega t + \phi)$

Maximum speed (at equilibrium, $x = 0$): $v_{max} = A\omega$

Acceleration

$a(t) = -A\omega^2\cos(\omega t + \phi) = -\omega^2 x$

Maximum acceleration (at amplitude, $x = \pm A$): $a_{max} = A\omega^2$

Key relationship: $a = -\omega^2 x$

Acceleration is always proportional to (and opposite in direction to) displacement!

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Mass-Spring System

For a mass $m$ attached to spring with constant $k$:

Period: $T = 2\pi\sqrt{\frac{m}{k}}$

Frequency: $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Key Observations:

• Heavier mass → longer period (slower oscillation)
• Stiffer spring (larger $k$) → shorter period (faster oscillation)
• Independent of amplitude! (as long as spring obeys Hooke's Law)

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Simple Pendulum

For a pendulum with length $L$ and small amplitude:

Period: $T = 2\pi\sqrt{\frac{L}{g}}$

Key Observations:

• Longer pendulum → longer period
• Independent of mass!
• Independent of amplitude! (for small angles < 15°)
• Only depends on length and gravity

Note: For large angles, period does depend on amplitude (not SHM anymore).

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⚠️ Common Mistakes

Mistake 1: Confusing Period and Frequency

• Period $T$: time for ONE oscillation (seconds)
• Frequency $f$: oscillations per second (Hz)
• They are reciprocals: $f = 1/T$

Mistake 2: Thinking Amplitude Affects Period

In ideal SHM, period is independent of amplitude!

• Large swing or small swing, same period
• As long as Hooke's Law holds

Mistake 3: Sign of Restoring Force

Force is ALWAYS opposite to displacement: $F = -kx$

• If $x > 0$ (stretched right), then $F < 0$ (force left)
• If $x < 0$ (compressed left), then $F > 0$ (force right)

Mistake 4: Pendulum Period and Mass

Pendulum period does NOT depend on mass!

• Only length $L$ and gravity $g$ matter
• Heavy or light bob, same period (if same length)

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Restoring Force

Definition: Force that always acts to return object to equilibrium

Characteristics:

• Always points toward equilibrium
• Proportional to displacement (for SHM)
• Creates oscillatory motion

For Spring: $F = -kx$

For Pendulum (small angle $\theta$):

• Restoring force = component of weight tangent to arc
• $F = -mg\sin\theta \approx -mg\theta$ (for small $\theta$)
• For small angles: $\theta \approx \frac{x}{L}$
• So: $F \approx -\frac{mg}{L}x$ (proportional to $x$!)

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Problem-Solving Strategy

For Mass-Spring Problems:

1. Identify: mass $m$, spring constant $k$
2. Calculate period: $T = 2\pi\sqrt{\frac{m}{k}}$
3. Calculate frequency: $f = 1/T$
4. Find amplitude: from initial conditions
5. Maximum speed: $v_{max} = A\omega$
6. Maximum acceleration: $a_{max} = A\omega^2$

For Pendulum Problems:

1. Identify: length $L$
2. Check: small angle approximation valid? ($\theta < 15°$)
3. Calculate period: $T = 2\pi\sqrt{\frac{L}{g}}$
4. Note: independent of mass and (small) amplitude

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Applications

Clocks and Timekeeping

• Pendulum clocks: invented by Christiaan Huygens (1656)
• Period independent of amplitude (for small swings)
• Reliable timekeeping

Seismographs

• Detect earthquake vibrations
• Record ground oscillations
• Mass-spring system measures ground motion

Musical Instruments

• Vibrating strings (guitars, violins)
• Vibrating air columns (flutes, organ pipes)
• Vibrating membranes (drums)

Molecular Vibrations

• Atoms in molecules vibrate about equilibrium
• Absorption spectra reveal molecular structure
• Chemical bond strength ~ spring constant

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Key Formulas Summary

| Quantity | Symbol | Mass-Spring | Pendulum | |----------|--------|-------------|----------| | Restoring Force | $F$ | $-kx$ | $-mg\sin\theta \approx -mg\theta$ | | Period | $T$ | $2\pi\sqrt{\frac{m}{k}}$ | $2\pi\sqrt{\frac{L}{g}}$ | | Frequency | $f$ | $\frac{1}{2\pi}\sqrt{\frac{k}{m}}$ | $\frac{1}{2\pi}\sqrt{\frac{g}{L}}$ | | Angular Frequency | $\omega$ | $\sqrt{\frac{k}{m}}$ | $\sqrt{\frac{g}{L}}$ | | Max Speed | $v_{max}$ | $A\omega$ | $A\omega$ | | Max Acceleration | $a_{max}$ | $A\omega^2$ | $A\omega^2$ |

Universal: $f = 1/T$, $\omega = 2\pi f$, $a = -\omega^2 x$