On constraint forces and d'Alembert's Principle

In summary, according to d'Alembert's Principle, the virtual work done by constraint forces must be zero. However, this principle is not always true when considering friction, which arises from velocity-dependent constraints. It is also unclear if friction can be considered a constraint force or an applied force. When restricting the principle to holonomic constraints, it is intuitively true that the constraint forces must be perpendicular to the constraint surface at each point at any fixed moment in time. However, the net work of the constraint forces on an object moving on a sphere with an accelerating path is not zero, but the virtual work is zero. It is uncertain if this principle can be proven from Newton's Laws and what a counter-example would look like
  • #1
Boorglar
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According to d'Alembert's Principle, the virtual work done by constraint forces must be zero.

I have a few things needing to be clarified. First, as we know from friction, d'Alembert's Principle is not always true (friction usually does work, and is not normal to the constraint surface). On the other hand, friction arises from velocity-dependent constraints (such as no-slip conditions). In fact, I am not even sure if friction can be considered a constraint force, since we could add it to the "applied forces".

Now, if we restrict d'Alembert's Principle to forces arising from holonomic constraints (constraints which do not depend on the velocities), it seems intuitively true. In that case, the principle says that the constraint forces must be perpendicular to the constraint surface at each point at any fixed moment in time. For example, if the constraint is [itex](x-t^2)^2 + y^2 + z^2 = l^2[/itex] (path lies on a sphere accelerating in the x direction) then the normal force at each point is "obviously" perpendicular to the sphere (directed towards its center [itex](t^2, 0, 0)[/itex]).
In this example, the net work of the constraint forces on an object moving on the sphere is not zero (the sphere is accelerating) but the virtual work is zero (according to d'Alembert's Principle).

Please let me know if what I said is not correct (or could be said better).

Is d'Alembert's Principle, restricted to holonomic constraints, provable from Newton's Laws?
And if it is not, what would a counter-example look like (if any is known)?

Also, what are some non-holonomic constraints which agree with d'Alembert's Principle?
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
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1. What is d'Alembert's Principle?

D'Alembert's Principle is a fundamental principle in classical mechanics that states that the sum of the external forces and the inertial forces acting on a system in equilibrium is equal to zero. This principle is used to analyze the motion of systems subject to constraint forces.

2. What are constraint forces?

Constraint forces are forces that arise due to the constraints imposed on a system. These constraints can be either external, such as a fixed support or a rigid surface, or internal, such as tension in a string or compression in a spring. Constraint forces are perpendicular to the constraints and do not contribute to the work done on the system.

3. How is d'Alembert's Principle used in mechanics?

D'Alembert's Principle is used to analyze the motion of systems in equilibrium, where the sum of the external forces and the inertial forces acting on the system is equal to zero. This allows us to determine the unknown forces and accelerations in the system and predict its motion.

4. What is the difference between internal and external constraints?

External constraints are imposed on a system by external objects, such as walls or supports, and do not affect the internal structure of the system. Internal constraints, on the other hand, are forces that arise within the system itself, such as tension in a string or compression in a spring, and can affect the overall motion of the system.

5. Are there any limitations to d'Alembert's Principle?

D'Alembert's Principle is based on the assumption that the system is in equilibrium, which means that the net force and net torque acting on the system are both zero. This may not hold true for systems that are not in equilibrium, and in such cases, other principles and laws must be applied to analyze the motion of the system.

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