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Simple Pendulum

Simple Pendulum

A simple pendulum consists of a small diameter ball suspended from a string, which has negligible mass but is strong enough to not stretch. In our daily life, pendulums have many uses, such as in clocks, on a swing set, and on a sinker on a fishing line.
The period of a simple pendulum depends on two factors: its length and the acceleration due to gravity. The period is completely independent of any other factors, such as mass or maximum displacement. For small displacements, a pendulum...

Physical Pendulum

Physical Pendulum

When a rigid body is hanging freely from a fixed pivot point and is displaced, it oscillates similar to a simple pendulum and is known as a physical pendulum. The period and angular frequency of a physical pendulum are obtained by using the small-angle approximation and drawing parallels with a spring-mass system. The small-angle approximation (sinθ=θ) is valid up to about 14°.
When dealing with complicated systems, the mass moment of inertia is an important parameter, as it...

Torsional Pendulum

Torsional Pendulum

A torsional pendulum involves the oscillation of a rigid body in which the restoring force is provided by the torsion in the string from which the rigid body is suspended. Ideally, the string should be massless; practically, its mass is much smaller than the rigid body's mass and is neglected.
As long as the rigid body's angular displacement is small, its oscillation can be modeled as a linear angular oscillation. The amplitude of the oscillation is an angle. The role of mass is played...

Measuring Acceleration Due to Gravity

Measuring Acceleration Due to Gravity

Consider a coffee mug hanging on a hook in a pantry. If the mug gets knocked, it oscillates back and forth like a pendulum until the oscillations die out.
A simple pendulum can be described as a point mass and a string. Meanwhile, a physical pendulum is any object whose oscillations are similar to a simple pendulum, but cannot be modeled as a point mass on a string because its mass is distributed over a larger area. The behavior of a physical pendulum can be modeled using the principles of...

Forced Oscillations

Forced Oscillations

When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...

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Related Experiment Video

Updated: Sep 8, 2025

Author Spotlight: Enhancing Engineering Education via WebVR-Based Online Laboratories

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Published on: February 23, 2024

Machine-learning potential of a single pendulum.

Swarnendu Mandal¹, Sudeshna Sinha², Manish Dev Shrimali¹

¹Central University of Rajasthan, Ajmer, Rajasthan, India 305817.

Physical Review. E

|June 16, 2022

Summary

This summary is machine-generated.

A single driven pendulum can perform complex learning tasks, demonstrating reservoir computing potential. This minimal system leverages rich dynamics, offering efficient machine learning applications.

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Area of Science:

Computational neuroscience
Nonlinear dynamics
Machine learning

Background:

Reservoir computing typically uses high-dimensional systems.
Physical systems can serve as computational substrates.
Transient dynamics in nonlinear systems are often underutilized.

Purpose of the Study:

To investigate a single driven pendulum as a low-dimensional reservoir for computing.
To explore the use of transient dynamics in a single system for learning tasks.
To demonstrate the feasibility of a minimal one-node reservoir for efficient machine learning.

Main Methods:

Numerical simulations of a driven pendulum.
Proof-of-principle experimental realization.
Analysis of temporal and non-temporal data processing tasks.

Main Results:

A single driven pendulum successfully performed learning tasks.
The system demonstrated accuracy and robustness in data processing.
Transient dynamics of the pendulum proved to be a valuable computational resource.

Conclusions:

A single, low-dimensional system can effectively implement reservoir computing.
This approach offers an efficient alternative for designing reservoir layers.
The study highlights the significant machine learning potential of simple nonlinear systems.