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 ( ω t + φ ) θ(t) = θ_0 \cos(\omega t + φ) θ ( t ) = θ 0 cos ( ω t + φ )
θ 0 θ_0 θ 0 represents initial angular displacement
ω \omega ω denotes angular frequency, related to period by ω = 2 π / T \omega = 2π/T ω = 2 π / T
Angular velocity equation
Represents rate of change of angular position
Obtained by differentiating angular displacement equation
For small angles: ω ( t ) = − ω θ 0 sin ( ω t + φ ) \omega(t) = -\omega θ_0 \sin(\omega t + φ) ω ( t ) = − ω θ 0 sin ( ω 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 ) = − ω 2 θ 0 cos ( ω t + φ ) α(t) = -\omega^2 θ_0 \cos(\omega t + φ) α ( t ) = − ω 2 θ 0 cos ( ω 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 = m g cos θ + m l ω 2 T = mg \cos θ + m l \omega^2 T = m g cos θ + m l ω 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 = − m g sin θ F = -mg \sin θ F = − m g sin θ (exact) or F ≈ − m g θ F ≈ -mg θ F ≈ − m g 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 = m g h = m g l ( 1 − cos θ ) U = mgh = mgl(1 - \cos θ) U = m g h = m g l ( 1 − cos θ ) where h is height above lowest point
Approximated as U ≈ 1 2 m g l θ 2 U ≈ \frac{1}{2}mg l θ^2 U ≈ 2 1 m g 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 = 1 2 m v 2 = 1 2 m l 2 ω 2 K = \frac{1}{2}mv^2 = \frac{1}{2}ml^2 \omega^2 K = 2 1 m v 2 = 2 1 m l 2 ω 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 k m 2\sqrt{km} 2 km
Decay of amplitude
Exponential decay for viscous damping (A ( t ) = A 0 e − γ t A(t) = A_0 e^{-γt} A ( t ) = A 0 e − γ t )
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 = 1 2 π g l f_r = \frac{1}{2π}\sqrt{\frac{g}{l}} f r = 2 π 1 l g
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: d 2 θ d t 2 + g l θ = 0 \frac{d^2θ}{dt^2} + \frac{g}{l}θ = 0 d t 2 d 2 θ + l g θ = 0
Full nonlinear equation: d 2 θ d t 2 + g l sin θ = 0 \frac{d^2θ}{dt^2} + \frac{g}{l}\sin θ = 0 d t 2 d 2 θ + l g sin θ = 0
Additional terms added for damping and driving forces
Small angle solution
Yields simple harmonic motion solution
Period given by T = 2 π l g T = 2π\sqrt{\frac{l}{g}} T = 2 π g l
Angular frequency ω = g l \omega = \sqrt{\frac{g}{l}} ω = l g
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 e q = I M d l_{eq} = \frac{I}{M d} l e q = M d I 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