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Work and Energy for Variable Forces01:10

Work and Energy for Variable Forces

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When an object is acted upon by a variable force, the amount of work done and the change in energy of the object can be more complex to calculate compared to when a constant force is applied. Work is the product of force and displacement, while energy is the capacity of a system to do work. When a constant force is applied to an object, the work done can be calculated as the product of the force and the distance moved in the direction of the force. However, when a variable force is applied, the...
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Moment of a Force: Problem Solving01:29

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Understanding the scalar formulation of the moment of a force and applying it correctly through problem-solving is crucial in designing and analyzing mechanical systems. Here are the steps for problem-solving with the moment of a force:
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Moment of a Force: Scalar Formulation01:18

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The moment of a force, also known as torque, measures the ability of the force to create rotational motion in a body about an axis. It is a vector quantity, meaning it has both magnitude and direction. This concept is used extensively in engineering, physics, and mechanics.
Consider a simple example of a flywheel being rotated about a point, O, by applying a force to it. In this case, the moment arm is the perpendicular distance between the point O and the line of action of the force. The...
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Moment of a Force: Vector Formulation01:27

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The moment of force refers to the measure of the rotational tendency of a force. It occurs when a force is applied in such a way that it produces a twisting or rotational motion rather than linear motion. The moment arm of a force is the perpendicular distance from the line of action of the force to the axis of rotation. The moment of force is not a scalar but a vector quantity.
The vector formulation of the moment of force is the cross-product of the position and force vectors. The...
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Moment of a Force About an Axis: Scalar01:28

Moment of a Force About an Axis: Scalar

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The moment of a force about an axis is a crucial concept in mechanics that helps determine an object's rotational motion around a specific point or axis. The moment of force can be calculated using scalar analysis, which involves considering the perpendicular distance between the axis of rotation and the line of action of the force or simply the moment arm.
To better understand the concept of moment of force, consider the example of a cyclist riding a bicycle. When the cyclist applies force on...
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Moment of a Force About an Axis: Vector01:29

Moment of a Force About an Axis: Vector

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When a force is exerted on an object, it can cause that object to rotate about an axis. The moment of a force, also known as torque, measures the force's ability to cause that rotation. In the case of a cyclist pedaling a bicycle, the force exerted on the pedal causes the crankshaft to rotate, which in turn causes the wheel to spin. The moment of the force exerted on the pedal drives the wheel's rotation.
First, establish a coordinate system to understand how the moment of a force...
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Updated: Jan 29, 2026

Ensemble Force Spectroscopy by Shear Forces
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Protocol Dependence and State Variables in the Force-Moment Ensemble.

Ephraim S Bililign1, Jonathan E Kollmer1, Karen E Daniels1

  • 1Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA.

Physical Review Letters
|February 9, 2019
PubMed
Summary

Researchers explored statistical mechanics for granular materials using photoelastic experiments. They identified two key state variables, angoricity and keramicity, crucial for understanding granular material behavior under stress.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Materials Science

Background:

  • Equations of state for granular materials are often developed using stress-based ensembles.
  • Temperature-like variables are incorporated into these ensembles to model granular material behavior.

Purpose of the Study:

  • To experimentally validate the efficacy of stress-based ensembles with temperature-like variables for granular materials.
  • To identify and characterize state variables governing the mechanical behavior of granular systems.

Main Methods:

  • Experiments were conducted on a 2D photoelastic granular system.
  • Three distinct loading conditions were applied: uniaxial compression, biaxial compression, and simple shear.
  • Interparticle forces and force-moment tensors were analyzed.

Main Results:

  • The normal component of the force-moment tensor follows an exponential distribution, while the deviatoric component is Gaussian.
  • Two state variables were identified: tensorial angoricity (α̂) and keramicity (κ).
  • Both variables are inversely proportional to confining pressure, but only keramicity demonstrates protocol independence.

Conclusions:

  • The findings imply that statistical mechanics for granular materials should conserve the force-moment tensor and force-tiling area.
  • Keramicity (κ) emerges as a valid state variable for granular materials, whereas angoricity (α̂) acts as a process variable.