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Example 5

When a spring of natural length L is compressed a distance x it exerts a force F = -kx. The negative sign indicates that the force is in the opposite direction from x (Figure 8.6.3).

08_exp-log_functions-342.gif

Figure 8.6.3

When x is negative the spring is expanded and the equation F = -kx still holds.

Suppose a mass m is attached to the end of the spring and at time t = 0 is at position x0 and has velocity v0. The motion of the mass follows the differential equation

08_exp-log_functions-343.gif

The general solution is

x = a cos cot + b sin ωt where 08_exp-log_functions-344.gif Using the initial conditions, the motion of the mass is

08_exp-log_functions-345.gif

This function is periodic with period 2π/ω, so as expected the mass oscillates back and forth.

In the following second order equation, hyperbolic sines and cosines arise. The general solution of the differential equation

d2y/dx2 = y

is

y = a cosh x + b sinh x.

We see that cosh x and sinh x are solutions because

08_exp-log_functions-346.gif, 08_exp-log_functions-347.gif,

08_exp-log_functions-348.gif, 08_exp-log_functions-349.gif.

Another solution is ex. Note that

08_exp-log_functions-350.gif


Last Update: 2006-11-16