Harmonic oscillation in classical mechanics

[sya deela]

Homework Statement

An object of mass m = 300g is attached to a spring with a constant k = 3.0Nm-1 and is at rest on a smooth horizontal floor in a fluid where the resistive force is assumed to be linearly proportional to the velocity v. the object is then displaced 10mm to the right of the equilibrium position and released. Given the constant of proportionality c1 = 0.2kgs-1;

Homework Equations

i) sketch the free body diagram (FBD) and write the equation of motion for the object immediately after it is set in motion
ii) write the solution to equation of motion in i) and describe the motion, and
iii) sketch the graph of the position versus time for the object.

The Attempt at a Solution

i) F(x) = -kx
Fv-F(x)= m dv/dt
-c1v-kx= m dv/dt

ii) -c1v-kx= m dv/dt after inegrate both sides v0 is zero
then v=-kx/c1(1-e(-c1t/m))

i can't get the equation for position versus time...

 

[eys_physics]

Use $v=dx/dt$.

 

[sya deela]

Use $v=dx/dt$.

i already try it..but as i know that for harmonic oscillation the graph should be cos / sin .So i don't know how to relate it.

 

[eys_physics]

Homework Statement

An object of mass m = 300g is attached to a spring with a constant k = 3.0Nm-1 and is at rest on a smooth horizontal floor in a fluid where the resistive force is assumed to be linearly proportional to the velocity v. the object is then displaced 10mm to the right of the equilibrium position and released. Given the constant of proportionality c1 = 0.2kgs-1;

Homework Equations

i) sketch the free body diagram (FBD) and write the equation of motion for the object immediately after it is set in motion
ii) write the solution to equation of motion in i) and describe the motion, and
iii) sketch the graph of the position versus time for the object.

The Attempt at a Solution

i) F(x) = -kx
Fv-F(x)= m dv/dt
-c1v-kx= m dv/dt

ii) -c1v-kx= m dv/dt after inegrate both sides v0 is zero
then v=-kx/c1(1-e(-c1t/m))

i can't get the equation for position versus time...

I don't understand what you are doing at ii). Here both $x$ and $t$ depends on $t$. So, your equation is
$$-c1 v(t)-kx(t)=mdv/dt$$
By using $v(t)=dx/dt$ in this equation, you can derive a second-order differential equation for $x(t)$.