a spring arranged parallel to a rough ramp inclined at 30 degrees is used to sto

Question

a spring arranged parallel to a rough ramp inclined at 30 degrees is used to stop packages being sent down the ramp. the spring is initially uncompressed and had a spring constant of 1500 lbs/ft. a 200 pound package is sent down the ramp with an initial velocity of 8 ft/s at a distance of 25 feet away from the spring measured along the ramps surface. the coefficient of kinetic friction between the ramp and the package is 0.2. use potential energy concepts to find the maximum compression of the spring.

Explanation / Answer

let the mass of the block be m
resolve the forces
along the direction of motion force is m.g.sin30
along the normal, force is m.g.cos30
let the compression in spring be x

the situation when block just comes to rest,
the force of spring and friction act in opposite direction to the motion
the force balance yields:

1500x + 0.2m.g.cos30 = m.g.sin30
put m=200lbm
g=32.7ft/s2

1500x = 200.g (sin30 - 0.2cos30)

x= (2g/15)*(0.5 - 0.1732)

x=1.425 ft

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