It Can Be Shown That . . .

(1")     .

Also recall Equation 5" of the Spring Constant, k it can be shown that page:

(2")     .

and Equation 6":

(3")     .

Integrating Equation 3" once yields:

(4")     

where

C1 is the constant of integration.

Substitute in the boundary conditions:

(5")     

and

(6")     

yields:

(7")     .

Inserting Equation 7" into Equation 2" yields:

(8")     

which reduces to:

(9")     .

Substitute Equation 9" into Equation 1" yields:

(10")    

which reduces to:

(11")    .

Rearranging Equation 7" of the Spring Constant, k it can be shown that page, which relates tip displacement to the end force, yields:

(12")     .

Substituting Equation 12" into Equation 11" yields:

(1)      .