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Related Concept Videos

Linear Momentum

Linear Momentum

The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum implies a tendency to continue on course—to move in the same direction; we tend to speak of sports teams or politicians gaining and maintaining the momentum to win. Momentum is also associated with great mass and speed and is often considered when talking about collisions. For example, when rugby players collide and fall to the ground, their...

Moment-of-Momentum Equation

Moment-of-Momentum Equation

The moment-of-momentum equation is a critical tool for analyzing the torque produced by the rotating blades of a wind turbine. This equation is derived by applying Newton's second law to a fluid particle, which states that the rate of change of linear momentum is equal to the external force acting on the particle.

Principle of Linear Impulse and Momentum for a Single Particle

Principle of Linear Impulse and Momentum for a Single Particle

Linear momentum is a fundamental concept in physics that describes the motion of an object. It is a vector quantity, having a magnitude equal to the product of its mass and its velocity, and direction along the object's velocity. On the other hand, linear impulse, also known as momentum impulse, is a concept in physics related to the change in the linear momentum of an object. Impulse is a vector quantity defined as the product of force and the time over which the force is applied.
Delving into...

Impulse-Momentum Theorem

Impulse-Momentum Theorem

The total change in the motion of an object is proportional to the total force vector acting on it and the time over which it acts. This product is called impulse, a vector quantity with the same direction as the total force acting on the object.
By writing Newton's second law of motion in terms of the momentum of an object and the external force acting on it, and simultaneously using the definition of the impulse vector, it can be shown that the total impulse on an object is equal to its net...

Application of the Linear Momentum Equation

Application of the Linear Momentum Equation

The application of the linear momentum equation can be used to analyze the forces needed to hold a 180-degree pipe bend in place with flowing water. In this case, water flows through the bend with a constant cross-sectional area of 0.01 square meters and a flow velocity of 15 meters per second. The pressure at the entrance is 0.2 Megapascals and the pressure at the exit is 0.16 Megapascals.
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Conservation of Linear Momentum for a System of Particles

Conservation of Linear Momentum for a System of Particles

In the dynamic realm of billiards, a fascinating interplay of forces governs the motion of cue balls and stationary balls. When the cue ball collides with a stationary ball, linear momentum is exchanged. The cue ball imparts a fraction of its linear momentum to the stationary ball, causing the cue ball to decelerate while initiating the motion of the stationary ball.
The impulsive force at play during this interaction is of extremely short duration, rendering its impulse negligible. When...

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Related Experiment Videos

Analysis of the back-propagation algorithm with momentum.

V V Phansalkar¹, P S Sastry

¹Dept. of Electr. Eng., Indian Inst. of Sci., Bangalore.

IEEE Transactions on Neural Networks

|January 1, 1994

Summary

The back-propagation algorithm with momentum stabilizes local minima for least squares error. Other equilibrium points in this neural network training method are unstable.

Area of Science:

Computer Science
Artificial Intelligence
Machine Learning

Background:

The back-propagation algorithm is a cornerstone of training artificial neural networks.
Understanding the stability of equilibrium points is crucial for effective network training and convergence.

Purpose of the Study:

To analyze the stability of equilibrium points in the back-propagation algorithm when incorporating a momentum term.
To investigate the behavior of local minima within the sum of least squares error landscape.

Main Methods:

Mathematical analysis of the back-propagation algorithm with momentum.
Lyapunov stability analysis applied to equilibrium points.

Main Results:

Demonstration that all local minima of the sum of least squares error are stable under the analyzed conditions.

Related Experiment Videos

Identification of other equilibrium points as unstable.

Conclusions:

The inclusion of a momentum term in the back-propagation algorithm ensures the stability of local minima for sum of least squares error.
Unstable equilibrium points may lead to divergence or oscillations during training, highlighting the importance of stable minima.