Questions 2 help A 8-kg mass is attached to a linear spring(k=800N/m) that is st

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Questions 2 help

A 8-kg mass is attached to a linear spring(k=800N/m) that is stretched 25 cm from its neutral position. The mass is released from the rest. Determine the general relationship between the initial energy stored in the spring, and the energy dissipated by friction. If the plane is frictionless, determine the maximum velocity of the mass. If the coefficient of kinetic friction is 0.3, determine the maximum velocity of the mass. If the coefficient of the kinetic friction is 0.3, determine the maximum compression the of the spring.

Explanation / Answer

a) Ei + W friction = Ef

Ei - Energy loss friction = KE + PE spring

b) if no friction Ei = Ef
1/2 kx^2 = 1/2 mv^2
0.5*800*.25^2 = 0.5*8*v^2
v=2.5 m/s

c) E friction = u m g (A-x)

1/2 kA^2 - u m g (A-x) = KE + 1/2 k x^2
KE = 1/2 kA^2 - u m g (A-x) - 1/2 kx^2
to find max KE
dKE/dx= u m g - k x = 0
x = u mg /k = 0.3*8*9.81/800
so
0.5*8*v^2 = 0.5*800*.25^2 - 0.3*8*9.81*(.25-0.3*8*9.81/800) - 0.5*800*(0.3*8*9.81/800)^2
v=2.206 m/s
d) max compression when v = 0
1/2 kA^2 - um g (A-x) = 1/2 kx^2
0.5*800*.25^2 - 0.3*8*9.81*(.25-x) = 0.5*800*x^2
x=-0.191
so max compression is 0.191 m

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