2+2=2x+2
-2 to both sides
2=2x
Divide 2
X=1
1/2x3/4=3/12
3/12 simplified is 1/4
Answer:
The y-intercept
Step-by-step explanation:
x-2y=10
-2y=10-x
y=x-5
Use Hooke's law to find the spring constant. If it takes 8N to stretch the spring by 0.1m, then
![8\text{ N}=k(0.1\text{ m})\implies k=80\dfrac{\text N}{\text m}](https://tex.z-dn.net/?f=8%5Ctext%7B%20N%7D%3Dk%280.1%5Ctext%7B%20m%7D%29%5Cimplies%20k%3D80%5Cdfrac%7B%5Ctext%20N%7D%7B%5Ctext%20m%7D)
I'm going to assume the spring is fixed to a ceiling, and that any stretching in the downward direction counts as movement in the positive direction. The spring's motion is then modeled by
![y''(t)+80y(t)=0](https://tex.z-dn.net/?f=y%27%27%28t%29%2B80y%28t%29%3D0)
where
![y(t)](https://tex.z-dn.net/?f=y%28t%29)
is the position of the spring's free end as it moves up and down. Solving this is easy enough: the characteristic solution will be
![y(t)=C_1\cos4\sqrt5t+C_2\sin4\sqrt5t](https://tex.z-dn.net/?f=y%28t%29%3DC_1%5Ccos4%5Csqrt5t%2BC_2%5Csin4%5Csqrt5t)
Given that the spring is stretched to a length of 1m (a difference of 0.25m from its natural length), and is released with no external pushing or pulling, we have the two initial conditions
![y(0)=\dfrac14](https://tex.z-dn.net/?f=y%280%29%3D%5Cdfrac14)
and
![y'(0)=0](https://tex.z-dn.net/?f=y%27%280%29%3D0)
.
![y(0)=\dfrac14\implies\dfrac14=C_1\cos0+C_2\sin0\implies C_1=\dfrac14](https://tex.z-dn.net/?f=y%280%29%3D%5Cdfrac14%5Cimplies%5Cdfrac14%3DC_1%5Ccos0%2BC_2%5Csin0%5Cimplies%20C_1%3D%5Cdfrac14)
![y'(0)=0\implies 0=-4\sqrt5C_1\sin0+4\sqrt5C_2\cos0\implies C_2=0](https://tex.z-dn.net/?f=y%27%280%29%3D0%5Cimplies%200%3D-4%5Csqrt5C_1%5Csin0%2B4%5Csqrt5C_2%5Ccos0%5Cimplies%20C_2%3D0)
So the spring's motion is dictated by the function