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Kinematic Equations: Problem Solving01:15

Kinematic Equations: Problem Solving

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When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the appropriate kinematic equations to solve for the unknowns. Either one or two kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities. Generally, the number of equations required is the same as the number of unknown quantities in the given example. Two-body pursuit problems always require two...
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Linear Approximation in Time Domain01:21

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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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First Law: Particles in Two-dimensional Equilibrium01:18

First Law: Particles in Two-dimensional Equilibrium

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Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration; but it is important to remember that these conditions are relative. For instance, an object may be at rest when viewed from one frame of reference, but that same object would appear to be in motion when viewed by someone moving at a constant velocity.
Newton's first law tells us about...
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Equation of Motion: General Plane motion - Problem Solving01:16

Equation of Motion: General Plane motion - Problem Solving

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Consider a lawn roller with a mass of 100 kg, a radius of 0.2 meters, and a radius of gyration of 0.15 meters. A force of 200 N is applied to this roller, angled at 60 degrees from the horizontal plane. What will be the angular acceleration of the lawn roller?
The friction between the roller and the ground is characterized by two coefficients. The static friction coefficient is 0.15, while the kinetic friction coefficient is 0.1. These values are crucial in understanding the interaction between...
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Equations of Equilibrium in Three Dimensions01:30

Equations of Equilibrium in Three Dimensions

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When analyzing structures or systems at rest, it is necessary to ensure they are in equilibrium. This is where the vector and scalar equations of equilibrium come into play. These equations are crucial in ensuring a structure is stable and will not collapse or fall apart. The vector and scalar equations of equilibrium provide a framework for analyzing the forces acting on a body.
According to the vector equations of equilibrium, the vector sum of all the external forces acting on a body must...
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Kinematic Equations - II01:17

Kinematic Equations - II

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The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant velocity, and the distance traveled due to a change in velocity. Similar to the first kinematic equation, this equation is also only valid when the acceleration is constant throughout the motion of an object.
Suppose a car merges into freeway traffic on a 200 m long ramp. If its initial velocity is 10 m/s and it accelerates at 2 m/s2, then the...
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Related Experiment Video

Updated: Jun 5, 2025

A Human-machine-interface Integrating Low-cost Sensors with a Neuromuscular Electrical Stimulation System for Post-stroke Balance Rehabilitation
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Balance equations for physics-informed machine learning.

Sandor M Molnar1, Joseph Godfrey2, Binyang Song3

  • 1Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan, Republic of China.

Heliyon
|December 10, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces a novel framework using balance equations to systematically construct physics-informed machine learning (PIML) residual loss terms. This method ensures ML models adhere to physical laws across diverse scientific and engineering domains.

Keywords:
Balance equationsComputational methodologiesElasticityElectrodynamicsFluid dynamicsMachine learningPhysics-informed systemsThermodynamics

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Area of Science:

  • Physics-informed machine learning (PIML)
  • Computational science
  • Applied mathematics

Background:

  • Traditional machine learning (ML) models may violate physical laws.
  • Physics-based models constrain ML using physical laws via loss terms.
  • Deriving complex differential equations for physical laws in ML is challenging.

Purpose of the Study:

  • To propose a systematic framework for constructing residual loss terms in PIML.
  • To ensure ML solutions are consistent with the laws of physics.
  • To advance PIML development across multiple domains.

Main Methods:

  • Developed a new framework based on balance equations for PIML.
  • Proposed a unified approach to derive differential equations from a generic balance equation.
  • Demonstrated practical application with worked examples.

Main Results:

  • The balance equation method provides a systematic approach to formulating residual loss terms.
  • A unified framework can incorporate physical laws into ML models.
  • This method ensures physical integrity in partial differential equation (PDE) solutions.

Conclusions:

  • The proposed balance equation method offers a generalized approach to PIML.
  • This framework unifies the treatment of fundamental physical equations.
  • It may lead to more efficient development of physics-based ML for complex systems.