# Linearly Damped Harmonic Motion with Sliding Friction

In Problem 2/54, I have come up with a solution only one decimal place off from the books results. I am posting to see if someone can look over my solution to see if it is correct.

To begin the equation given:

a = u*g - (k/m)x

Only works in cases where it only moves in a direction where the friction force is strictly positive (i.e. the block is moving leftward).

To solve the case of motion in the opposite direction must be considered, the equation for motion opposite this direction is:

a = -u*g - (k/m)x

Now using the relationship in the book:

Integrating and noting that each starting and stopping point has zero velocity:

(1) 0 = u*g(xf - xi) -(k/2m)(xf^2 - xi^2)

(2) 0 = -u*g(xf - xi) -(k/2m)(xf^2 - xi^2)

Now using (1) the first stop start point occurrs at:

xf = .23866m

On the way back however the motion is reveresed and equation (2) is used to find:

xf = .022666m

The solution in the book was: 23.3mm

Did I make a mistake or why is there such a seemingly slight difference in the answer?