# Homework Help: Simple Harmonic Motion and friction

1. Dec 8, 2009

### henryc09

1. The problem statement, all variables and given/known data
Mass m moves along a line on a rough table and is attached on either side to a stretched spring (both have equal spring constants k). The coefficients of static and sliding friction between mass and the table are equal with value $$\mu$$

(a) show in the abscence of friciton the particle executes SHM with angular frequency$$\omega$$ $$\sqrt{2k/m}$$

(b) Now include the effect of friction. Suppose particle is released at time t=0 with positive displacement Xo from equilibrium. Describe initial motion when (i)2kXo> $$\mu$$mg and (ii) when 2kXo =< $$\mu$$mg

(c) For case (i) write down the equation of satisfied by the displacement x of the mass as long as it remains moving. Verify it is satisfied by a solution of the form:

x(t)=Acos(wt) + Bsin(wt) + C

and find the values of A, B and C for the data given.

(d) Find the time t1 and position x1 at which the particle next comes to rest.

2. Relevant equations

3. The attempt at a solution

Been a while since I last did S.H.M, but so far I have without friction total force on object will be -2kx which is equal to -mw2x so i can show the first bit.

For part b I said that when 2kXo > $$\mu$$mg it will oscillate with lower frequency and it's amplitude will gradually decrease, when 2kX0 =< $$\mu$$mg it will remain stationary.

Part C is where I'm stuck. I think -mw2x= - 2kx + $$\mu$$mg but not sure how to show the solution and also how to go about working out the constants. Any point in the right direction would be appreciated.