Spring connected to a block problem

[Zetor]

"Spring connected to a block" problem

A block has an initial velocity of V and is connected to a damper and a spring like this:

http://prism2.mem.drexel.edu/~rares/MassSpringDamper.jpg

The problem is to figure how far the spring vill maximally be compressed.

The damping is linearily proportional to the velocity of the block. It is possible to solve it with more or less standardised differential equations, I however want to try a different approach.

I want to solve by doing an energy equation like this:

mv^2/2 = kx^2/2 + $\int cx' dx$

where x is the maximum compression and the integral the force from the damper integrated over the distance x. The speed over time is denoted as the derivate of x which equals speed. The constant C has such dimension that V(t)*c=F(t).
The second term is the potential energy stored in the spring.

However, since I don't know x' some trick needs to be done. Can this problem be solved by this approach?

 

[eljose79]

Thank you for sharing your approach to solving this problem. It is always interesting to see different methods being used to tackle a problem. Your energy equation is a valid approach to solving this problem, but there are a few things that need to be considered.

First, the damping force is not simply proportional to the velocity of the block, but also to the velocity of the spring. This means that your integral should also include the velocity of the spring, not just the block.

Second, the constant C is not dimensionless, as you have stated. It actually has the dimensions of [force]/[velocity], which is consistent with your statement that V(t)*c=F(t). This is important to keep in mind when setting up your energy equation.

Lastly, the trick you mentioned is known as the principle of virtual work, where the integral of the damping force is replaced with the virtual work done by the damping force. This allows for the use of the principle of conservation of energy, and is a common approach in solving problems involving springs and dampers.

In conclusion, your approach is valid, but it is important to consider the factors mentioned above. I encourage you to continue exploring different methods and techniques in problem solving, as it can lead to a deeper understanding of the concepts involved. Best of luck in solving this problem!