Answer:
a) The mass moves a distance of 0.625 m up the slope before coming to rest
b) The distance moved by the mass when it is connected to the spring is 0.6 m
c) ![\mu = 0.206](https://tex.z-dn.net/?f=%5Cmu%20%3D%200.206)
Explanation:
Spring constant, k = 70 N/m
Compression, x = 0.50 m
Mass placed at the free end, m = 2.2 kg
angle, θ = 41°
Potential Energy stored in the spring, ![PE= 0.5 kx^2](https://tex.z-dn.net/?f=PE%3D%200.5%20kx%5E2)
![PE = 0.5 * 70 * 0.5^2\\PE = 8.75 J](https://tex.z-dn.net/?f=PE%20%3D%200.5%20%2A%2070%20%2A%200.5%5E2%5C%5CPE%20%3D%208.75%20J)
According to the principle of energy conservation
PE = mgh
8.75 = 2.2 * 9.8 * h
h = 0.41
If the mass moves a distance d from the spring
sin 41 = h/d
sin 41 = 0.41/d
d = 0.41/(sin 41)
d = 0.625 m
The mass moves a distance of 0.625 m up the slope before coming to rest
b) If the mass is attached to the spring
According to energy conservation principle:
Initial PE of spring = Final PE of spring + PE of block
![0.5kx_1^2 = 0.5kx_2^2 + mgh\\x_2 = d - x_1 = d - 0.5\\h = d sin 41\\0.5*70*0.5^2 = 0.5*70*(d-0.5)^2 + 2.2*9.8*d*sin41\\8.75 = 35(d^2 - d + 0.25) + 14.15d\\8.75 = 35d^2 - 35d + 8.75 + 14.15d\\35d^2 = 20.85d\\d = 0.6 m](https://tex.z-dn.net/?f=0.5kx_1%5E2%20%3D%200.5kx_2%5E2%20%2B%20mgh%5C%5Cx_2%20%3D%20d%20-%20x_1%20%3D%20d%20-%200.5%5C%5Ch%20%3D%20d%20sin%2041%5C%5C0.5%2A70%2A0.5%5E2%20%3D%200.5%2A70%2A%28d-0.5%29%5E2%20%2B%202.2%2A9.8%2Ad%2Asin41%5C%5C8.75%20%3D%2035%28d%5E2%20-%20d%20%2B%200.25%29%20%2B%2014.15d%5C%5C8.75%20%3D%2035d%5E2%20-%2035d%20%2B%208.75%20%2B%2014.15d%5C%5C35d%5E2%20%3D%2020.85d%5C%5Cd%20%3D%200.6%20m)
The distance moved by the mass when it is connected to the spring is 0.6 m
3) The spring potential is converted to increased PE and work within the system.
mgh = Fd + 0.5kx²...........(1)
d = x
, h = dsinθ
kinetic friction force
, F = μmgcosθ
mgdsinθ + μmg(cosθ)d = 0.5kd²
mgsinθ + μmgcosθ = 0.5kd
sinθ + μcosθ = kd/(2mg)
![\mu = \frac{\frac{kd}{2mg} - sin\theta}{cos\theta} \\\\\mu = \frac{\frac{70*0.5}{2*2.2*9.8} - sin41}{cos41} \\\\\mu = 0.206](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%5Cfrac%7B%5Cfrac%7Bkd%7D%7B2mg%7D%20-%20sin%5Ctheta%7D%7Bcos%5Ctheta%7D%20%5C%5C%5C%5C%5Cmu%20%3D%20%5Cfrac%7B%5Cfrac%7B70%2A0.5%7D%7B2%2A2.2%2A9.8%7D%20-%20sin41%7D%7Bcos41%7D%20%5C%5C%5C%5C%5Cmu%20%3D%200.206)