A spring arranged parallel to a rough ramp inclined at 27 degree, is used to sto

Question

A spring arranged parallel to a rough ramp inclined at 27 degree, is used to stop packages being sent down that ramp. The spring is initially uncompressed, and has a spring constant of 1500 lbs/ft. A 161 pound package is sent down the ramp with an initial velocity of 8.2 ft/s at a distance of 25.4 feet away from the spring measured along the ramps surface, the coefficient of kinetic friction between the ramp and package is 0.22. Use work-kinetic energy methods to find the maximum compression of the spring after interacting with the sliding package.

Explanation / Answer

initial elastic potential energy Uie = 0

final kinetic energy of the system Kf = 0

final gravitational potential energy Ufg =

final elastic potential energy Ufe = (1/2)*k*x^2

normal force n = uk*m*g*costheta

frictional force fk = uk*n = uk*m*g*costheta

work done by friction = -fk*L = -uk*m*g*costheta*(L+x)

final elastic potential energy Uf = (1/2)*k*x^2

from work - kinetic energy

Wg + We + Wf = dK

Uig - Ufg + Uie - Ufe + Wf = Kf - Ki

m*g*sintheta*(L+x) - (1/2)*k*x^2 - uk*m*g*costheta*(L+x) = -(1/2)*m*vi^2

(161*32*sin27*(25.4+x)) - ((1/2)*1500*x^2) - (0.22*161*32*cos27*(25.4+x)) = -(1/2)*161*8.2^2

x = 8.16 ft

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