Mathematical Tools for Physics

4—Differential Equations 84

the spring is to a good approximation proportional to the distance that the mass has moved from its equilibrium
position.
If the coordinatexis measured from the mass’s equilibrium position, the equationF~=m~asays

x

m

d^2 x

dt^2

=−kx (1)

If there’s friction (and there’salways friction), the force has another term. Now how do you describe friction
mathematically? The common model for dry friction is that the magnitude of the force is independent of the
magnitude of the mass’s velocity and opposite to the direction of the velocity. If you try to write that down in a
compact mathematical form you get something like

F~friction=−μkFN~v

|~v|

(2)

This is a mess to work with. It can be done, but I’m going to do something different. See problem 31 however.
Wet friction is easier to handle mathematically because when you lubricate a surface, the friction becomes velocity
dependent in a way that is, for low speeds, proportional to the velocity.

F~friction=−b~v (3)

Neither of these two representations is a completely accurate description of the way friction works. That’s far
more complex than either of these simple models, but these approximations are good enough for many purposes
and I’ll settle for them.
Assume “wet friction” and the differential equation for the motion ofmis

m

d^2 x

dt^2

=−kx−b

dx

dt

(4)

This is a second order, linear, homogeneous, differential equation, which simply means that the highest derivative
present is the second, the sum of two solutions is a solution, and a constant multiple of a solution is a solution.
That the coefficients are constants makes this an easy equation to solve.