Maybe I should have posted this question in 'elementary physics' but the question seemed rather difficult to me(although involving elementary concepts) so i am posting it at this forum.Here goes.

A block of mass m is attached to a spring of spring constant k.the other end of the spring is attached to a wall.The mass rests on a surface with friction coefficient [tex]\mu[/tex].Now the mass is given a displacement [tex]x_{0}[/tex] from its equilibrium position.Calculate x at any subsequent time.

The forces acting on the block are(taking the equilibrium position as origin)

force due to the spring:

[tex]\\F_{spring} = -kxi\hat \\

[/tex]

[tex]

\\ F_{friction} = \mu mgi\hat \\ \mbox{ when the block is moving in the -i direction}

[/tex]

[tex]

F_{friction} = -\mu mgi\hat \\ \mbox{ when the block is moving in +i direction}

[/tex]

The equations i obtain are

[tex] m\ddot x +kx = \mu mg\ \mbox{when moving towards -i} \\

[/tex]

[tex]

\\ \mbox{and} m\ddot x + kx = -\mu mg\ \mbox{when moving towards +i}

[/tex]

Solving them i get,for the first half cycle:

[tex]

\\x = (x_{0} - \frac{\mu mg}{k})\cos \frac{k}{m}t + \frac{\mu mg}{k}\\

[/tex]

[tex]

\mbox{At the end of the first half cycle,the position of the block is,then } x'_{0} = -x_{o} + 2\frac{\mu mg}{k}

[/tex]

[tex]

\\ \mbox{Now,if i take t= 0 at} x= x'_{0} \ \mbox{ ,i get,from solving the second differential equation}[/tex]

[tex]

\\x = -( x'_{o}+ \frac{\mu mg}{k}) \cos \frac{k}{m}t - \frac{\mu mg}{k}

[/tex]

[tex]

\mbox{At the end of this half-cycle the position is }

[/tex]

[tex]

x_{0} - 4\frac{\mu mg}{k}

[/tex]

So i'm getting the result that for each half cycle the distance traversed goes down by [tex] 2\frac{\mu mg}{k}[/tex].I get the same reult from work-energy theorem.

is my method correct?if yes,is there a more elegant way of doing it?is there at least an elegant way of writing the answer?In my answer i'm putting t=0 at the start of each half-cycle,so i can't boil it down to less than 2 equations.

P.S:this is te first time i'm using latex.

**1. The problem statement, all variables and given/known data**A block of mass m is attached to a spring of spring constant k.the other end of the spring is attached to a wall.The mass rests on a surface with friction coefficient [tex]\mu[/tex].Now the mass is given a displacement [tex]x_{0}[/tex] from its equilibrium position.Calculate x at any subsequent time.

**2. Relevant equations**The forces acting on the block are(taking the equilibrium position as origin)

force due to the spring:

[tex]\\F_{spring} = -kxi\hat \\

[/tex]

[tex]

\\ F_{friction} = \mu mgi\hat \\ \mbox{ when the block is moving in the -i direction}

[/tex]

[tex]

F_{friction} = -\mu mgi\hat \\ \mbox{ when the block is moving in +i direction}

[/tex]

**3. The attempt at a solution**The equations i obtain are

[tex] m\ddot x +kx = \mu mg\ \mbox{when moving towards -i} \\

[/tex]

[tex]

\\ \mbox{and} m\ddot x + kx = -\mu mg\ \mbox{when moving towards +i}

[/tex]

Solving them i get,for the first half cycle:

[tex]

\\x = (x_{0} - \frac{\mu mg}{k})\cos \frac{k}{m}t + \frac{\mu mg}{k}\\

[/tex]

[tex]

\mbox{At the end of the first half cycle,the position of the block is,then } x'_{0} = -x_{o} + 2\frac{\mu mg}{k}

[/tex]

[tex]

\\ \mbox{Now,if i take t= 0 at} x= x'_{0} \ \mbox{ ,i get,from solving the second differential equation}[/tex]

[tex]

\\x = -( x'_{o}+ \frac{\mu mg}{k}) \cos \frac{k}{m}t - \frac{\mu mg}{k}

[/tex]

[tex]

\mbox{At the end of this half-cycle the position is }

[/tex]

[tex]

x_{0} - 4\frac{\mu mg}{k}

[/tex]

So i'm getting the result that for each half cycle the distance traversed goes down by [tex] 2\frac{\mu mg}{k}[/tex].I get the same reult from work-energy theorem.

is my method correct?if yes,is there a more elegant way of doing it?is there at least an elegant way of writing the answer?In my answer i'm putting t=0 at the start of each half-cycle,so i can't boil it down to less than 2 equations.

P.S:this is te first time i'm using latex.

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