Phy 319 -- HW#2

We want to first numerically solve the differential equations we found in class for  the generalized incldine plane with friction.  We had two coupled differential equations in the x- and z-directions.  

Below is a set of input lines, the first few of which input some given constants.

ClearAll[]

g = 9.8
muk = 0.3
\(\(\(theta = 30.  Degree\)\(
\)
    \)\)

Here we solve the diff. Eq. for x(t) and z(t), remembering to put in the  initial conditions (all of which are zero except the initial z-velocity).

nsolution  =  
      NDSolve[{x''[t] == 
            g*Sin[theta]  -  
              muk*g*Cos[theta]*x'[t]/
                  Sqrt[x'[t]^2 + z'[t]^2],  z''[
              t] == -muk*g*
              Cos[theta]*z'[t]/Sqrt[x'[t]^2 + z'[t]^2],  
          x[0] == z[0] == x'[0] == 0, z'[0] == 0.2}, {x, z}, {t, 0, 
          2}]

Now let's plot with axes the x-motion vs. the z-motion.

Plot[Evaluate[x[t] /. nsolution], {t, 0, 0.25}]

p1 = 
      ParametricPlot[Evaluate[{z[t], x[t]} /. nsolution], {t, 0, 0.25},  
        AxesLabel ->  {"z,accros", "x,down"}]

This looks reasonable -- intially the block moves in the z-direction, but gravity pulls it down the plane in the x-direction.  Friction modifies this.  

Now let's plot the approximate solution from class.  Note that v_z(0)/v(0) = 1 and v_x(0)=0 for our initial conditions.

g = 9.8;
muk = 0.3;
theta = 30.  Degree;
p2 = 
      ParametricPlot[{-g/2*muk*Cos[theta]*t^2 + 0.2*t, 
          g/2*Sin[theta]*t^2}, {t, 0, 0.25}]

Wow!  That friction force overwelms gravity in a hurry...very unphysical in a short time.
Let's over lay them.

Show[p1, p2]

Now look at shorter times.  They agree much better, but don't fully converge.  The problem is likely in where we make an additional approximation in the equations for DeltaXnot'' and DeltaZnot''.  We not only say that X''>>DeltaX.  In the expression for DeltaX and DeltaZ we also let Xnot'(t) = Xnot'(0).  This threw away important early-time information.  So, our approximation is solvable, but not terribly accurate.  You can see how numerical techniques have really opened the door to analyzing complex motion (and this problem isn't really all that complex!)

Show[
      ParametricPlot[Evaluate[{z[t], x[t]} /. nsolution], {t, 0, 0.05},  
        AxesLabel ->  {"z,accros", "x,down"}], 
      ParametricPlot[{-g/2*muk*Cos[theta]*t^2 + 0.2*t, 
          g/2*(Sin[theta] - muk*Cos[theta])*t^2}, {t, 0, 0.05}]]