Answer:
Compression distance: ![d \approx 0.102\,m](https://tex.z-dn.net/?f=d%20%5Capprox%200.102%5C%2Cm)
Explanation:
According to this statement, we know that system is non-conservative due to the rough patch. By Principle of Energy Conservation and Work-Energy Theorem, we have the following expression that represents the system having a translational kinetic energy (
), in joules, at the expense of elastic potential energy (
), in joules, and overcoming work losses due to friction (
), in joules:
(1)
By definitions of translational kinetic and elastic potential energies and work losses due to friction, we expand the equation described above:
(2)
Where:
- Mass of the block, in kilograms.
- Final velocity of the block, in meters per second.
- KInetic coefficient of friction, no unit.
- Gravitational acceleration, in meters per square second.
- Width of the rough patch, in meters.
- Spring constant, in newtons per meter.
- Compression distance, in meters.
If we know that
,
,
,
,
and
, then the compression distance of the spring is:
![\frac{1}{2}\cdot m \cdot v^{2} +\mu\cdot m\cdot g \cdot s = \frac{1}{2} \cdot k \cdot d^{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20m%20%5Ccdot%20v%5E%7B2%7D%20%2B%5Cmu%5Ccdot%20m%5Ccdot%20g%20%5Ccdot%20s%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%20%5Ccdot%20k%20%5Ccdot%20d%5E%7B2%7D)
![m\cdot v^{2} + 2\cdot m\cdot g \cdot s = k\cdot d^{2}](https://tex.z-dn.net/?f=m%5Ccdot%20v%5E%7B2%7D%20%2B%202%5Ccdot%20m%5Ccdot%20g%20%5Ccdot%20s%20%3D%20k%5Ccdot%20d%5E%7B2%7D)
![d = \sqrt{\frac{m\cdot (v^{2}+2\cdot g\cdot s)}{k} }](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%5Cfrac%7Bm%5Ccdot%20%28v%5E%7B2%7D%2B2%5Ccdot%20g%5Ccdot%20s%29%7D%7Bk%7D%20%7D)
![d \approx 0.102\,m](https://tex.z-dn.net/?f=d%20%5Capprox%200.102%5C%2Cm)