Answer:
the mass drop by 6.5cm before coming to rest.
Explanation:
Given that:
the mass of the block M= 1.40 kg
angle of inclination θ = 30°
spring constant K = 40.0 N/m
mass of the suspended block m = 60.0 g = 0.06 kg
initial downward speed = 1.40 m/s
The objective is to determine how far does it drop before coming to rest?
Let assume it drops at y from a certain point in the vertical direction;
Then :
Workdone by gravity on the mass of the block is:
[tex]w_1g = 1.4*9.81*y*sin30[/tex] = - 6.867y
Workdone by gravity on the mass of the suspended block is:
[tex]w_2g = 0.06*9.81[/tex] = 0.5886
The workdone by the spring = -1/2ky²
= -0.5 × 40y²
= -20 y²
The net workdone is = -20 y² - 6.867y + 0.5886y
According to the work energy theorem
Net work done = Δ K.E
-20 y² - 6.867y + 0.5886 = 1/2 × 0.06 × 1.4²
-20 y² - 6.867y + 0.5886 = 0.5 × 0.1176
-20 y² - 6.867y + 0.5886 = 0.0588
-20 y² - 6.867y + 0.5886 - 0.0588
-20 y² - 6.867y + 0.5298 = 0
multiplying through by (-)
20 y² + 6.867y - 0.5298 = 0
Using the quadratic formula:
[tex]\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]
where;
a = 20 ; b = 6.867 c= - 0.5298
[tex]\dfrac{-(6.867) \pm \sqrt{(6.867)^2-(4*20*-0.5298)}}{2*(20)}[/tex]
[tex]= \dfrac{-(6.867) + \sqrt{(6.867)^2-(4*20*-0.5298)}}{2*(20)} \ \ \ OR \ \ \ \dfrac{-(6.867) - \sqrt{(6.867)^2-(4*20*-0.5298)}}{2*(20)}[/tex]= 0.0649 OR −0.408
We go by the positive integer
y = 0.0649 m
y = 6.5 cm
Therefore; the mass drop by 6.5cm before coming to rest.
