A package delivery service (we can\'t say any names but their initials are F and

Question

A package delivery service (we can't say any names but their initials are F and E), has asked your help in designing a system to bring packages that have been sliding along delivery chutes before they get loaded onto the trucks. The packages, with a mass of 6 kg, are stationary at the top and then they slide down a chute that is 5.5-m-high. The delivery chute has rollers along it which make it virtually frictionless. The delivery chute makes and angle of 21.1 degrees. A stopping spring is located at the very end of the entire chute but the 3.2-m-wide horizontal slowing chute between the bottom of the delivery chute and right in front of the spring does not have rollers and is not frictionless (mu_k = 0.28). The stopping spring has a spring constant of 450 N/m. For simplicity, assume the chute under the spring is frictionless, otherwise the math gets ugly.

What is the speed of the package at the bottom of the chute?

What is the speed of the box just before hitting the stopping spring?

How far is the stopping spring compressed as it bring the package to a stop?

Explanation / Answer

1) PE at top of chute = mgh = 6*9.8*5.5 = 323.4 J

This PE is converted to KE at the bottom of chute

So 0.5*m*v2 = 323.4

so v = 10.38 m/s

2) kinetic friction = 0.28*6*9.8 = 16.46 N

distence of stopping chute = 3.2m

so work done by friction = 16.46*3.2 = 52.68 J

Energy left = 323.4-52.68 = 270.71 J

So velocity = 9.499 m/s

3) This energy is stored in spring

0.5*k*x*x = 270.11

So x = 1.09 m

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