Moving Constraints: Virtual Work in Classical Mechanics

In summary, the virtual work done in the case of moving constraints is not zero, as shown in classical mechanics by R. Douglas Gregory on page 345 under the heading of Lagrange's equations with moving constraints. However, the virtual work done by constraint forces is still considered zero even in time-dependent cases, which may seem confusing. This is because when taking the dot product with the virtual velocity, only the partial derivative of the position vector r_i with respect to the generalized coordinate q_j is considered, not the time derivative. This may be difficult to understand, but hopefully, with further information or rewording, it can be clarified.
  • #1
Sumit Dey
1
0
the virtual work done in the case of moving constraints is obviously not zero(argument as shown in classical mechanics by r douglas gregory page 345 under the heading of Lagrange's equations with moving constraints.i just wanted to understand how come the virtual work done the constraint forces in still zero even in the time dependant cases. why is he just taking into consideration of the partial derivative of the position vector r_i wrt to the generalized coord q_ j only and not the time derivative while taking the dot product with the virtual velocity.. please help!
 
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  • #2
Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

1. What is virtual work in classical mechanics?

Virtual work in classical mechanics is a mathematical concept used to analyze the motion of a system under the influence of external forces. It involves calculating the work done by these forces on a virtual displacement of the system, which is an imaginary displacement that satisfies the constraints of the system but does not change its position.

2. How is virtual work used to solve problems in classical mechanics?

Virtual work is used to determine the equilibrium conditions and equations of motion for a system, making it a useful tool for solving problems in classical mechanics. By setting the virtual work done by the external forces equal to zero, we can derive equations that describe the behavior of the system.

3. What are the advantages of using virtual work in classical mechanics?

Virtual work allows us to analyze complex systems with multiple forces and constraints in a simplified manner. It also enables us to consider non-conservative forces, such as friction, in our calculations. Additionally, virtual work can be used to solve problems involving virtual displacements, which are easier to visualize and understand compared to real displacements.

4. Are there any limitations to using virtual work in classical mechanics?

One limitation of virtual work is that it only applies to systems in static equilibrium or undergoing small displacements. It also assumes that the constraints of the system are holonomic, meaning they can be described by a set of equations involving only the coordinates of the system. Furthermore, virtual work does not take into account the effects of inertia and is therefore not suitable for analyzing systems with high acceleration or rotation.

5. How is virtual work related to the principle of least action?

The principle of least action states that a system will always follow the path that minimizes the action, which is the integral of the Lagrangian function over time. This principle is closely related to virtual work, as the equations derived from virtual work can also be used to obtain the equations of motion through the principle of least action. This connection allows us to use virtual work to solve problems in both statics and dynamics in classical mechanics.

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