# A 2-kg small block is dropped from rest. The spring has a constant k = 500 N/m. If the entire...

## Question:

A 2-kg small block is dropped from rest. The spring has a constant k = 500 N/m. If the entire track is frictionless except for the 1 m between points B and C, where the coefficient of kinetic friction is 0.15, find the maximum compression of the spring.

## Conservation Of Energy:

The energy cannot be created and cannot be destroyed and will be converted from one form into another form in the conversion process for example potential energy contained within the system can be converted into the kinetic energy when the object is allowed to fall from a certain height.

Given

Mass of the small block is {eq}m= 2\ kg {/eq}

Spring constant is {eq}k= 500\ N/m {/eq}

Distance for which the track is having friction is {eq}1\ m {/eq}

Now for the potential energy at the top of the block:

Assuming the height of the track to be 5 m:

{eq}P.E=mgh\\ P.E=2\times 9.81\times 5\\ P.E=98.1\ J {/eq}

Now for the frictional energy loss:

{eq}F.E=\mu (mg)\times x\\ F.E=0.15\times (2\times 9.81)\times 1\\ F.E=2.943\ J {/eq}

Now for the Energy available at the spring will be :

{eq}\Delta E= P.E-F.E\\ \Delta E=98.1-2.943\\ \Delta E= 95.15\ J {/eq}

Now, this energy will be equal to spring potential energy :

{eq}95.15=\frac{1}{2}kx^2\\ 95.15=\frac{1}{2}(500)\times x^2\\ x=0.616\ m {/eq}

Thus, the maximum compression of the spring will be 0.616 m