Question

An 7.42 kg block drops straight down from a height of 1.34 m, striking a platform spring having a force constant of 1.10 103 N/m. Find the maximum compression of the spring.

Answer 1

First we must ask ourselves: what do we need to solve the problem? From where we place our reference system?

The first question is answered by the following: we must to find the impact block speed, and then we must determinate when that speed becomes 0, that is, when the spring fully stop movement. For the second question, we can set our reference system so that it matches the upper spring base.

We are going to solve this problem with a dynamic analysis, by the law of conservation of mechanical energy. it states that mechanical energy is equal to the variation of kinetic energy plus the variation of the potential energy, which in turn equals to the total work of the particle in its path.

  

now, in this case, exist a difference to considerate in the vatiation of potencial energy. This time we must consider the potential energy associated with spring. Our new variation of potential energy is given by:


Now, we know that kinetic energy is given by:  

As the speed in the inicial point is equals 0, and so is the final speed, our variation of kinetic energy is equal 0 too. Then, as the total mechanical energy is equal to 0 because there are only conservative forces in the fall of the mass, we have that:



Our final potencial energy is equals 0 because there is no heigh. Finally, we have that:

where:

we can find the maximum compression of the spring following the next equation as:

we can then divide the problem into two parts: 
a)from the starting point to the top of the spring.
b)from the top of the spring to its compression.

a) the total mechanical energy is equal to 0 because there are only conservative forces in the fall of the mass , then we have:

where the subscript "f" represents final and the subscript "i" represents initial. In the upper base of the spring, the final kinetic energy (Kf) is not equals 0, but the final potential energy (Uf) is equals 0 because, according to our reference system, there is no height. We also know that the initial kinetic energy Ki is also equal to 0 because the inicial block speed is equals 0. Then, we can find the final speed as:

Where:

Then we have that:   

b) now we will raise the problem from the time the block touches the spring until the spring stop block. For this we can raise the above equation of energy conservation, with the difference that this time it will consider the potential energy associated with spring . Our new variation of potential energy is given by:

Then, our new equation is given by:

Where the speed in the new initial kinetic energy is equal to the speed of the final kinetic energy in the previous a) point, plus, our new Kf becomes 0 because when the spring stops the block, its final speed becomes 0. Also, our inicial potencial energy Ui is equals 0