A 2000kg car begins from rest and rolls 50.0m down a frictionless 10 degrees incline. If there is a horizontal spring at the end of the incline, what spring constant is required to stop the car in a distance of 1.00m?

Answer 1

The car has lost potential energy the spring must absorb.

So the change height is 51sin10, and the mgh change of PE is then
2000*9.8*51sin10. But the trick here is stopping the car. THe energy must be absorbed by the spring, spring energy is 1/2 k 1^1, set them equal and

1/2 k=2000*9.8*51sin10 solve for k.
k=2*2000*9.8*51sin10

Answer 2

A book is placed on a horizontal wooden plane that is undergoing simple
harmonic motion with an amplitude of 1.0 m. The coefficient of friction
between the book and the horizontal wooden plane is given by micro = 0.5.
Determine the frequency of the horizontal wooden platform when the book
is about to slip from the horizontal wooden plane.