Deriving SHM: Help Needed for Challenging Problem in Textbook

[tonykoh1116]

This is actually the problem in the textbook.
I'm trying to derive the harmonic oscillator differential equation for this system, but It seems like it's very very challenging.

could anyone help me out?

Following is the question and figure from the textbook.

Homework Statement

A block of mass m is attached to a fixed support by a horizontal spring with force constant k and negligible mass. The block sits on a long horizontal board, with which it has coefficient of static friction μs and a smaller coefficient of kinetic friction μk. The board moves to the right at constant speed v. Assume the block spends most of its time sticking to the board and moving to the right, so the speed v is small in comparison to the average speed the blok has as it slips back toward the left.
Derive the harmonic oscillator differential equation for the system.

Homework Equations

F=ma
F(sp)=-kx
E=K+U

The Attempt at a Solution

I couldn't figure out.



( The figure has a copyright from the textbook).
I will delete this post if it could cause the copyright problem.

 

[mfb]

Your differential equation is relevant when the block moves to the left. It is sliding (with friction) and accelerated by the spring. Can you calculate the total force which acts on the block as function of its position?