Question

A block of mass m is pushed against an ideal spring of constant of k, compressing it over a distance x with respect to its natural length. When the block is released, it moves up a rough ramp of inclination θ and coefficient of friction μ_k.What is the maximum distance (d) that the block travels up the incline? You MUST use conservation of energy to solve this problem. Epress your answers in term of m, g, k, μ_k and θ.

Answer 1

The PE of spring get converted to the potential energy at the ramp and frictional work done

from conservation of energy

PE(spring) = PE(gravity) + Wf

1/2 k x^2 = m g h + uk m g d cos(theta)

using trigonometry, h = d sin(theta)

1/2 k x^2 = m g d sin(theta) + uk m g d cos(theta)

1/2 k x^2 = d (m g sin(theta) + uk m g cos(theta))

d = k x^2/2 m g (sin(theta) + uk cos(theta))

d = k x^2/[2 mg (sin + k cos)]