Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
WorkSheet 15
Place the coordinate system for the position of the mass at the appropriate place. Be sure
that 0 is at the proper place and that the positive axis is pointing down.
Spring (natural length)
Equilibrium with mass
Motion with mass
Now use the F = ma equation to …nd the di¤erential equation which describes the position,
x(t); of the mass. Assume that the forces are gravity (use g); restoring force in the spring
(use k); a resistive force (use c); and an external force (use f (t)): The …rst short underline
on the left of the = is for a derivative. The short underlines on the right of the = are for +
or ¡ (i.e. recall that a force has magnitude and direction).
m
=
|
{z
Gravity
}
|
{z
Spring
}
|
{z
Resistance
}
|
{z
External
Now rewrite the above in the usual 2nd order DE format. Be sure to explain any simpli…cation
that you do.
}
Assuming that there is no external force, the di¤erential equation which describes the position
of the mass is
mx00 + cx0 + kx = 0
What is the characteristic equation for this di¤erential equation? Use the variable r in the
characteristic equation.
Use the quadratic equation to …nd the solutions to the characteristic equation. That is
§ p
r=
What condition on m; c; and k will give real solutions to the characteristic equation? Will
these r’s be positive or negative? Write down the general solution.
What condition on m; c; and k will give exactly one solution to the characteristic equation?
Will this r be positive or negative? Write down the general solution.
What condition on m; c; and k will give complex solutions to the characteristic equation?
Write these r’s in standard complex form, namely, a § bi: