Answer:
a) the distance up to which the object compress the spring is x = 0.378m
b),the distance up to which the object compress the spring is x= 0.359
Explanation:
Given that,
Mass , m = 1.6kg
spring at vertical height ,h = 1.05m
Spring constant ,k = 315 N/m
Air resistance force = 0.750N
Let assume that the lowest point is the origin
Therefore, initial displacement is
![y_i = 1.05 + x](https://tex.z-dn.net/?f=y_i%20%3D%201.05%20%2B%20x)
And final displacement is
![y_f = 0](https://tex.z-dn.net/?f=y_f%20%3D%200)
Apply the conservation of energy
![\frac{mv_i^2}{2} +\frac{kx_i^2}{2} +mgy_i=\frac{mv_f^2}{2} +\frac{kx^2}{2} +mgy_f](https://tex.z-dn.net/?f=%5Cfrac%7Bmv_i%5E2%7D%7B2%7D%20%2B%5Cfrac%7Bkx_i%5E2%7D%7B2%7D%20%2Bmgy_i%3D%5Cfrac%7Bmv_f%5E2%7D%7B2%7D%20%2B%5Cfrac%7Bkx%5E2%7D%7B2%7D%20%2Bmgy_f)
![\frac{m(0)^2}{2} +\frac{k(0)^2}{2} +mg(1.05+x)=\frac{m(0)^2}{2} +\frac{kx^2}{2} +mg(0)](https://tex.z-dn.net/?f=%5Cfrac%7Bm%280%29%5E2%7D%7B2%7D%20%2B%5Cfrac%7Bk%280%29%5E2%7D%7B2%7D%20%2Bmg%281.05%2Bx%29%3D%5Cfrac%7Bm%280%29%5E2%7D%7B2%7D%20%2B%5Cfrac%7Bkx%5E2%7D%7B2%7D%20%2Bmg%280%29)
![(1.60)(9.81)(1.05 + x)^2=\frac{(315)}{2} x^2\\\\16.4808+15.969x=157.5x^2](https://tex.z-dn.net/?f=%281.60%29%289.81%29%281.05%20%2B%20x%29%5E2%3D%5Cfrac%7B%28315%29%7D%7B2%7D%20x%5E2%5C%5C%5C%5C16.4808%2B15.969x%3D157.5x%5E2)
![x^2-0.1014x-0.1046=0](https://tex.z-dn.net/?f=x%5E2-0.1014x-0.1046%3D0)
solve the equation using quadratic equation
![x = \frac{-(-0.1014\pm \sqrt{(-0.1014)^2-4(1)(-1046)} )}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-%28-0.1014%5Cpm%20%5Csqrt%7B%28-0.1014%29%5E2-4%281%29%28-1046%29%7D%20%29%7D%7B2%7D)
![x = \frac{0.1014\pm\sqrt{0.42868} }{2} \\\\x = \frac{0.1014\pm0.6547}{2} \\\\x = \frac{0.1014+0.6547}{2} or \frac{0.1014-0.6547}{2} \\\\x = 0.3781 or -0.2767](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B0.1014%5Cpm%5Csqrt%7B0.42868%7D%20%7D%7B2%7D%20%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B0.1014%5Cpm0.6547%7D%7B2%7D%20%5C%5C%5C%5Cx%20%3D%20%5Cfrac%7B0.1014%2B0.6547%7D%7B2%7D%20or%20%5Cfrac%7B0.1014-0.6547%7D%7B2%7D%20%5C%5C%5C%5Cx%20%3D%200.3781%20or%20-0.2767)
Take the positive value
Hence , the distance up to which the object compress the spring is x = 0.378m
b)If the air resistance is taken into cosideration
therefore apply conservation of energy
mg(1.05 + x) - fd = kx²/2
mg(1.05 +x) - (0.750N) (1.05 + x) = kx²/2
((1.60× 9.81) - 0.750)(1.05 + x) = (315/2)x²
14.946 (1.05 + x) = 157.5x²
15.6933 + 14.946x = 157.5x²
x² - 0.0949x - 0.00996
solve the equation using quadratic equation
x = 0.359 or -0.264
Hence ,the distance up to which the object compress the spring is x= 0.359