Simple Harmonic Motion

To fully specify the initial conditions of a particle in simple harmonic motion, we need:

1. The initial displacement (x₀) from the equilibrium position
2. The initial velocity (v₀) at that position

These two parameters completely determine the subsequent motion of the particle, including:

• The amplitude of oscillation
• The phase constant
• The exact position and velocity at any future time

The equation of motion can then be written as: x = A·sin(ωt + δ)

where A and δ are determined from x₀ and v₀.

During a complete cycle of simple harmonic motion in a spring-mass system:

1. At the equilibrium position (x = 0):

   • All energy is kinetic energy (KE = ½mv²)
   • Potential energy equals zero
   • Velocity is maximum

2. At maximum displacement (x = ±A):

   • All energy is potential energy (PE = ½kA²)
   • Kinetic energy equals zero
   • Velocity equals zero

3. At intermediate positions:

   • Energy exists in both forms
   • Total energy remains constant: E = ½kA² = ½mω²A²

The continuous transformation between potential and kinetic energy occurs while maintaining constant total mechanical energy throughout the motion.

At extreme positions (x = ±A):

• Position: Maximum displacement from equilibrium
• Velocity: Zero (v = 0)
• Potential energy: Maximum (U = ½kA²)
• Kinetic energy: Zero (K = 0)
• Total energy: E = ½kA²

At equilibrium position (x = 0):

• Position: Zero displacement
• Velocity: Maximum (v_max = Aω)
• Potential energy: Zero (U = 0)
• Kinetic energy: Maximum (K = ½mv_max² = ½mA²ω²)
• Total energy: E = ½mA²ω²

Key relationship: ω = √(k/m), so the total energy expression remains constant (E = ½kA²) throughout the motion, showing energy conservation while the forms of energy interchange.

In a pendulum:

• The gravitational force (mg) resolves into two components:

  1. Tangential component: mg·sinθ (parallel to motion)
  2. Radial component: mg·cosθ (countered by tension)

• The tangential component mg·sinθ acts as the restoring force

• For small angles: sinθ ≈ θ (in radians)

• Therefore: restoring force ≈ -mg·θ

• Since θ = x/l (where x is arc displacement and l is length), this gives: F = -mg·(x/l) = -(mg/l)·x

• This follows Hooke's Law form F = -kx where k = mg/l

• The negative sign indicates the force opposes displacement

This tangential component is what creates the characteristic simple harmonic motion of a pendulum with period T = 2π√(l/g).

A torsional pendulum consists of:

• An extended body suspended by a wire or thread
• The body oscillates by rotating about the vertical axis
• Restoring torque is provided by the twisted wire/thread

The period is given by: $T = 2\pi\sqrt{\frac{I}{k}}$

Where:

• $I$ = moment of inertia of the body about the vertical axis
• $k$ = torsional constant of the wire

When the body is rotated through angle $\theta$:

• The wire exerts a restoring torque $\tau = -k\theta$
• The motion is simple harmonic for small amplitudes
• Angular acceleration is directly proportional to angular displacement

Analysis of forces on a physical pendulum:

1. Weight ($mg$): • Acts vertically downward through the center of mass • Magnitude = $mg$ • Creates torque of magnitude $mgl\sin\theta$ around pivot • This is the restoring torque that drives oscillation

2. Normal/contact force ($N$): • Acts at the pivot point • Balances other forces to maintain constraint • Contributes zero torque about the pivot point • Reason: force line of action passes through the axis of rotation

Key principle: Only forces with a moment arm relative to the pivot contribute to torque.

The torque equation becomes: $\tau = -mgl\sin\theta$

This analysis applies to any rigid body suspended from a fixed support, such as a circular ring on a nail or a rod suspended at a hole.

Note: While the normal force doesn't affect the rotational dynamics directly, it's essential for maintaining the pendulum's constraint.

When damping is reduced in a forced oscillation system:

• The peak amplitude at resonance increases significantly
• The resonance curve becomes narrower and sharper
• The system becomes more selective to frequencies near the natural frequency (ω₀)
• Energy transfer from the driving force to the oscillating system becomes more efficient

This explains why lightly damped systems (like wine glasses or bridges) can develop dangerously large amplitudes when driven at their resonant frequencies.

The vector method for combining multiple simple harmonic motions (SHMs) of the same frequency:

1. Representation: Each SHM is represented as a vector where:

   • Vector length = amplitude of the motion (A₁, A₂, etc.)
   • Vector direction = phase angle of the motion (δ₁, δ₂, etc.)

2. Vector Addition: These vectors are added tip-to-tail following vector addition rules

3. Resultant Motion: The final vector yields:

   • Magnitude (A) = resultant amplitude
   • Direction angle (ε) = resultant phase angle

4. Mathematical Expression: The combined motion is expressed as: $x = A\sin(\omega t + \varepsilon)$

5. For Two SHMs (e.g., x₁ = A₁sin(ωt) and x₂ = A₂sin(ωt + δ)):

   • Resultant amplitude: $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\delta}$
   • Resultant phase angle: $\varepsilon = \tan^{-1}\left(\frac{A_2\sin\delta}{A_1 + A_2\cos\delta}\right)$

This method can be extended to any number of SHMs by continuing the vector addition process for all component motions sharing the same angular frequency (ω).

Note: The key insight is that SHMs with the same frequency but different amplitudes and phases combine to form another SHM of the same frequency, but with a new amplitude and phase determined by vector addition.

When two perpendicular SHMs with the same frequency combine, they create specific paths (Lissajous figures) determined by:

1. Phase Difference (δ) - The critical shape determinant:

   • δ = 0: Straight line through origin with positive slope
   • δ = π: Straight line through origin with negative slope
   • δ = π/2: Ellipse with axes aligned with coordinate axes (circular if amplitudes equal)
   • Other values (0 < δ < π): Ellipse with rotated axes, direction depends on whether δ < π/2 or δ > π/2

2. Amplitude Relationship:

   • For straight lines (δ = 0): Slope = A₂/A₁
   • For ellipses: Determines eccentricity
   • Equal amplitudes with δ = π/2: Perfect circle

3. Mathematical Description:

   • General equation: x²/A₁² + y²/A₂² - 2xy·cosδ/(A₁A₂) = sin²δ
   • For straight line (δ = 0): y = (A₂/A₁)x
   • Combined amplitude (δ = 0): √(A₁² + A₂²)

4. Key Properties:

   • The particle oscillates with the same frequency as component motions
   • Traversal direction depends on sign of phase difference
   • When motions are in phase or 180° out of phase, the combined motion remains simple harmonic

When combining two SHMs of the same frequency in the same direction, both the amplitude and phase angle of the resultant motion can be determined using:

Resultant Amplitude: $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\delta}$

Resultant Phase Angle (relative to the first component): $\tan\varepsilon = \frac{A_2\sin\delta}{A_1 + A_2\cos\delta}$

Where:

• $A_1$, $A_2$ = individual amplitudes
• $\delta$ = phase difference between component motions
• $\varepsilon$ = phase angle of resultant motion relative to first component

Key Properties:

• Maximum amplitude ($A = A_1 + A_2$) occurs when $\delta = 0$ (in phase)
• Minimum amplitude ($A = |A_1 - A_2|$) occurs when $\delta = \pi$ (out of phase)
• For intermediate phase differences, amplitude varies between these extremes

This can be understood through vector representation, where the two SHMs are represented as vectors with the second vector at angle $\delta$ relative to the first, and the resultant vector having amplitude $A$ and phase angle $\varepsilon$.