An inclined plane of angle ? = 20.0 An inclined plane of angle ? = 20.0 degree h

Question

An inclined plane of angle ? = 20.0

An inclined plane of angle ? = 20.0 degree has a spring of force constant k = 465 N/m fastened securely at the bottom so that the spring is parallel to the surface as shown in the figure below. A block of mass m = 2.71 kg is placed on the plane at a distance d = 0.297 m from the spring. From this position, the block is projected downward toward the spring with speed v = 0.750 m/s. By what distance is the spring compressed when the block momentarily comes to rest?

Explanation / Answer

Let PE = 0 be defined at that instant when the block is just about to compress the spring. Denote by x the maximum compression of the spring. Then from the relation
(1) ..... sin(20) = h/x
where h is the vertical distance below PE = 0 where the maximum compression of the spring takes place, we find that h = x*sin(20). At the position of the maximum compression of the spring where the block momentarily comes to rest, we find that the total energy of the block is
(2) ..... Ef = ( 0.5 )*k*x^2 - m*g*x*sin(20)
If we take the total energy of the block at it's initial position, then we find that
(3) ..... Ei = (0.5)*m*(v^2) + m*g* [ d*sin(20) ]
Assuming no energy is lost during the whole process, conservation of energy applies, Ei = Ef, so that we get the relation
(4) ..... (0.5)*m*(v^2) + m*g* [ d*sin(20) ] = ( 0.5 )*k*x^2 - m*g*x*sin(20)
Substituting the values, we get the following results:
(5) ..... Ef = ( 0.5 )*( 465 )*x^2 - ( 2.39 )*( 9.8 )*x*( 0.3420) = (232.5)*(x^2) - (8.01)*x
(6) ..... Ei = (0.5)*(2.39)*(0.750)^2 + (2.39)*(9.8)*(0.330)*(0.3420) = 3.3156
Equivalently, (4) can be expressed as
(7) ..... (232.5)*(x^2) - (8.01)*x - 3.3156 = 0
which is a quadratic equation in x. Using the quadratic formula, we get
(8) ..... x1 = { - ( - 8.01) + sqroot [ ( - 8.01)^2 - 4*(232.5)* ( - 3.3156) ] } / [ 2*(232.5) ]
which gives x1 = 0.1479 m = 0.1480 m
(9) ..... x2 = { - ( - 8.01) - sqroot [ ( - 8.01)^2 - 4*(232.5)* ( - 3.3156) ] } / [ 2*(232.5) ]
which gives x2 = - 0.1034 m that is not an acceptable answer. The maximum compression of the spring will then be x1 = 0.1480 m

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