7.3 Pendulums

Pendulums are a classic example of harmonic motion in mechanics. They illustrate key concepts like periodic motion, energy conservation, and gravitational effects. Understanding pendulums provides a foundation for analyzing more complex oscillatory systems.

The simple pendulum model assumes ideal conditions, like a massless string and no friction. This simplification allows for mathematical analysis of the pendulum's motion, including equations for displacement, velocity, and acceleration. The small angle approximation further simplifies calculations for practical applications.

Simple pendulum model

• Fundamental concept in classical mechanics illustrating harmonic motion
• Serves as a simplified representation of more complex oscillatory systems
• Provides insights into periodic motion, energy conservation, and gravitational effects

Ideal pendulum assumptions

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• Massless, inextensible string supports a point mass
• No air resistance or friction affects the pendulum's motion
• Oscillations occur in a vertical plane with constant gravitational field
• Pivot point remains fixed and experiences no energy dissipation

Small angle approximation

• Assumes pendulum swings through small angles (typically less than 15 degrees)
• Allows simplification of trigonometric functions (sin θ ≈ θ)
• Leads to simple harmonic motion description with constant period
• Introduces error that increases with larger swing amplitudes

Period vs amplitude

• Period remains nearly constant for small amplitudes (isochronism)
• Slight increase in period observed for larger amplitudes
• Relationship described by elliptic integral for exact solution
• Amplitude dependence becomes significant for swings exceeding 20-30 degrees

Equations of motion

Angular displacement equation

• Describes position of pendulum as a function of time
• For small angles: $θ(t) = θ_0 \cos(\omega t + φ)$
• $θ_0$ represents initial angular displacement
• $\omega$ denotes angular frequency, related to period by $\omega = 2π/T$

Angular velocity equation

• Represents rate of change of angular position
• Obtained by differentiating angular displacement equation
• For small angles: $\omega(t) = -\omega θ_0 \sin(\omega t + φ)$
• Maximum angular velocity occurs at equilibrium position

Angular acceleration equation

• Describes rate of change of angular velocity
• Derived by differentiating angular velocity equation
• For small angles: $α(t) = -\omega^2 θ_0 \cos(\omega t + φ)$
• Proportional to angular displacement with opposite sign

Forces acting on pendulum

Tension in string

• Directed along the string towards the pivot point
• Magnitude varies throughout swing to maintain constant string length
• Reaches maximum at the bottom of swing, minimum at extremes
• Calculated using $T = mg \cos θ + m l \omega^2$

Gravitational force

• Constant downward force due to Earth's gravity
• Magnitude equals mass of bob multiplied by gravitational acceleration
• Resolved into components parallel and perpendicular to string
• Parallel component provides restoring force for oscillation

Restoring force

• Brings pendulum back towards equilibrium position
• Proportional to displacement for small angles (Hooke's law analogy)
• Given by $F = -mg \sin θ$ (exact) or $F ≈ -mg θ$ (small angle approximation)
• Responsible for simple harmonic motion behavior

Pendulum energy

Potential energy

• Maximum at extreme positions of swing
• Decreases as pendulum approaches equilibrium position
• Given by $U = mgh = mgl(1 - \cos θ)$ where h is height above lowest point
• Approximated as $U ≈ \frac{1}{2}mg l θ^2$ for small angles

Kinetic energy

• Maximum at equilibrium position (bottom of swing)
• Increases as pendulum moves away from extreme positions
• Calculated using $K = \frac{1}{2}mv^2 = \frac{1}{2}ml^2 \omega^2$
• Includes both translational and rotational components

Energy conservation

• Total mechanical energy remains constant in ideal pendulum
• Energy continuously converts between potential and kinetic forms
• Allows prediction of pendulum behavior at any point in oscillation
• Deviations from conservation indicate presence of non-conservative forces

Damped pendulums

Types of damping

• Viscous damping (proportional to velocity)
• Coulomb damping (constant frictional force)
• Hysteretic damping (internal material friction)
• Air resistance (combination of viscous and quadratic damping)

Damping coefficient

• Quantifies strength of damping force
• Determines rate of energy dissipation in system
• Influences decay rate of oscillation amplitude
• Critical damping occurs when coefficient equals $2\sqrt{km}$

Decay of amplitude

• Exponential decay for viscous damping ($A(t) = A_0 e^{-γt}$)
• Linear decay for Coulomb damping
• Logarithmic decrement measures rate of amplitude reduction
• Overdamped systems return to equilibrium without oscillation

Driven pendulums

Resonance frequency

• Frequency at which system response is maximized
• Occurs when driving frequency matches natural frequency of pendulum
• For undamped pendulum: $f_r = \frac{1}{2π}\sqrt{\frac{g}{l}}$
• Damping shifts resonance frequency slightly lower

Forced oscillations

• Result from external periodic force applied to pendulum
• Steady-state motion has frequency of driving force
• Amplitude and phase depend on driving frequency and damping
• Transient behavior occurs before steady-state is reached

Amplitude vs driving frequency

• Amplitude increases as driving frequency approaches resonance
• Peak amplitude occurs slightly below natural frequency for damped systems
• Amplitude decreases rapidly for frequencies above resonance
• Phase shift between driving force and pendulum motion varies with frequency

Applications of pendulums

Clocks and timekeeping

• Pendulum clocks use isochronous property for accurate timekeeping
• Escapement mechanism maintains pendulum oscillation
• Temperature compensation techniques improve accuracy (mercury pendulums)
• Largely superseded by quartz and atomic clocks for precision timekeeping

Seismometers

• Utilize pendulum principles to detect and measure ground motion
• Horizontal pendulums sense lateral earth movements
• Inverted pendulums used for vertical motion detection
• Modern seismometers often use electronic sensors instead of physical pendulums

Foucault pendulum

• Demonstrates Earth's rotation through precession of swing plane
• Period of precession depends on latitude (24 hours at poles, infinite at equator)
• Requires long pendulum and careful isolation from air currents
• Often displayed in science museums and universities

Mathematical analysis

Differential equations

• Pendulum motion described by nonlinear second-order differential equation
• For small angles: $\frac{d^2θ}{dt^2} + \frac{g}{l}θ = 0$
• Full nonlinear equation: $\frac{d^2θ}{dt^2} + \frac{g}{l}\sin θ = 0$
• Additional terms added for damping and driving forces

Small angle solution

• Yields simple harmonic motion solution
• Period given by $T = 2π\sqrt{\frac{l}{g}}$
• Angular frequency $\omega = \sqrt{\frac{g}{l}}$
• Solution accurate within 1% for angles up to about 23 degrees

Large angle behavior

• Requires numerical methods or series expansions for solution
• Period increases with amplitude (anharmonic oscillator)
• Can exhibit chaotic behavior for very large amplitudes or driven systems
• Elliptic integral solution provides exact period for any amplitude

Compound pendulums

Center of oscillation

• Point at which simple pendulum has same period as compound pendulum
• Located below center of mass for most shapes
• Distance from pivot to center of oscillation gives equivalent simple pendulum length
• Reversible pendulum uses this property to measure g accurately

Moment of inertia

• Measures resistance to rotational acceleration
• Depends on mass distribution relative to axis of rotation
• Affects period of compound pendulum oscillation
• Calculated using parallel axis theorem for complex shapes

Equivalent simple pendulum

• Simple pendulum with same period as compound pendulum
• Length determined by ratio of moment of inertia to static moment
• Given by $l_{eq} = \frac{I}{M d}$ where I is moment of inertia, M is mass, d is CM distance
• Allows application of simple pendulum formulas to compound systems

Experimental methods

Measuring period

• Use of stopwatch to time multiple oscillations for increased accuracy
• Photogate sensors for precise timing of pendulum passages
• Video analysis techniques for detailed motion study
• Importance of accounting for damping effects in long-duration measurements

Determining local gravity

• Reversible pendulum method for high-precision g measurements
• Kater's pendulum design minimizes errors from pivot friction
• Correction factors applied for air buoyancy, temperature, and latitude
• Modern absolute gravimeters achieve higher precision than pendulum methods

Error analysis

• Systematic errors from measuring length, mass, and time
• Random errors reduced through multiple measurements and statistical analysis
• Uncertainty propagation techniques applied to derived quantities
• Comparison of experimental results with theoretical predictions to validate models