I am having some trouble with conservation of energy

[kyin01]

Homework Statement

Homework Equations

$$K_{1}$$+$$U_{1}$$+$$W_{nc}$$=$$K_{2}$$+$$U_{2}$$

The Attempt at a Solution

Using the above equation

1/2m$$v^{2}$$ + 0 + xmg$$\mu$$ = 0 + 1/2k$$x^{2}$$

On the left side
When it hits the spring there is 0 potential because it has not yet compress the spring and it still remains on a horizontal surface. It has a kinetic energy and kinetic friction work
On the right side
I have 0 kinetic because it just finished compressing the spring and about to decompress so instead I get maximum potential energy.

So solving for K i get

$$\frac{mv^{2}+xmg\mu}{x^{2}}$$

But according to the answer it does not depend on the variable X (distance compressed by spring)
So I'm not entirely sure I know where I made my mistake

 

[PhanthomJay]

Aside from your algebra error and your incorrect assumption that the work done by friction is positive, you have neglected to incorporate the given condition that the object has no speed when the spring returns to its uncompressed length (that is, its speed is zero both when the spring is fully compressed, and when it returns to its uncompressed state). That will give you the info you need to eliminate the 'x' term. Remember that the work done by friction is negative.

 

[Moetasim]

, but I will try to provide some guidance on the concept of conservation of energy.

Conservation of energy is a fundamental principle in physics that states that energy cannot be created or destroyed, only transferred or converted from one form to another. In other words, the total energy of a closed system remains constant over time.

In the context of your problem, we can think of the system as the object being dropped onto the spring. Initially, the object has potential energy due to its position above the ground, and no kinetic energy. As it falls, it gains kinetic energy and loses potential energy. When it hits the spring, some of this kinetic energy is converted into potential energy stored in the spring as it compresses. However, there is also work being done by non-conservative forces, such as friction, which will decrease the amount of energy transferred to the spring.

After the object reaches its maximum compression and begins to decompress, the potential energy stored in the spring is converted back into kinetic energy. However, this time there is no work being done by non-conservative forces, so the amount of kinetic energy gained by the object is equal to the potential energy stored in the spring. This results in the total energy of the system remaining constant.

In your attempt at a solution, it seems like you may have mixed up some of the terms for potential and kinetic energy. Remember, potential energy is dependent on the position of an object in a force field, while kinetic energy is dependent on the object's motion. It may also be helpful to draw a free body diagram and carefully consider the forces acting on the object at different points in its motion.

Overall, it is important to carefully consider the different forms of energy in a system and how they are being transferred or converted in order to properly apply the principle of conservation of energy. I hope this helps guide you in the right direction in solving your problem.