Classical Mech doubt 4m Goldstein

[pardesi]

Note that there is no dependence of time, $$\delta t$$ is involved here( in the definition of virtual displacement that is $$\delta \vec{r_{i}}=\sum_{j} \frac{d \vec{r_{j}} \delta q_{j}}{d q_{j}}$$ ) ,since a virtual displacement by def depends only on the displacement of coordinates.(Only then is the virtual displacement perpendicular to the force of constraint if the constarint itself is changing with time.)

Can someone Prove the thing in bold letters?

 

[Valerie Park]

No, it is not possible to prove this statement. The statement is a definition of a virtual displacement, and as such cannot be proven.

 

[charng]

The statement in bold is a fundamental concept in classical mechanics, known as the principle of virtual work. It states that for a system in equilibrium, the virtual work done by the applied forces and the virtual work done by the constraint forces must be equal. In other words, the sum of the dot product of the applied forces and virtual displacements must be equal to the sum of the dot product of the constraint forces and virtual displacements.

To prove this, we can start with the definition of virtual displacement, which is a hypothetical displacement that satisfies the constraints of the system. This means that the virtual displacement is perpendicular to the constraint forces, as any component of the virtual displacement in the direction of the constraint forces would violate the constraints.

Next, we can consider the definition of virtual work, which is the dot product of the applied force and the virtual displacement. Since the virtual displacement is perpendicular to the constraint forces, the dot product of the constraint forces and virtual displacement will be zero. Therefore, the virtual work done by the constraint forces is also zero.

Now, if we apply the principle of virtual work, we can see that the sum of the virtual work done by the applied forces and the sum of the virtual work done by the constraint forces must be equal to zero. This means that the virtual work done by the applied forces must be equal and opposite to the virtual work done by the constraint forces.

In conclusion, the statement in bold can be proven by applying the principle of virtual work and considering the definitions of virtual displacement and virtual work. This principle is a fundamental concept in classical mechanics and is crucial in understanding the behavior of systems in equilibrium.