Explicit values of the Desired Functions.

The solution is also obtainable with a graphing calculator. Let f be f and q be q. Suppose that you calculate f as follows:
f = p - sin-1( r/L sin(q)).
IMPORTANT: Calculators return a value of arcsin between -90 and +90 degrees. Our values of interest lie between about 150 and 210 degrees. Therefore the adjustment must be made as shown.

Given next are the functions to be loaded.

The equation describing f

Input := 

e1 = Sin[f[t]] L == Sin[q[t]] r

Output =

L Sin[f[t]] == r Sin[q[t]]

The function describing f'

Input := 

e2 = D[e1,t] /. q'[t] -> w

Output =

L Cos[f[t]] f'[t] == r w Cos[q[t]]

Input := 

s2 = Solve[e2,f'[t]][[1,1]]

Output =

         r w Cos[q[t]] Sec[f[t]]
f'[t] -> -----------------------
                    L

The function describing f''

Input := 

e3 = D[e2,t]

Output =

                   2
-(L Sin[f[t]] f'[t] ) + L Cos[f[t]] f''[t] == 
 
  -(r w Sin[q[t]] q'[t])

Input := 

s3 = Solve[e3,f''[t]][[1,1]]

Output =

f''[t] -> -(
 
                                   2
     Sec[f[t]] (-(L Sin[f[t]] f'[t] ) + r w Sin[q[t]] q'[t])
     -------------------------------------------------------)
                                L

The function x

Input := 

e4 = X[t] == r Cos[w t] + L Cos[ f[t] ]

Output =

X[t] == r Cos[t w] + L Cos[f[t]]

The function x'

Input := 

e5 = D[e4,t]

Output =

X'[t] == -(r w Sin[t w]) - L Sin[f[t]] f'[t]

The function x''

Input := 

e6 = D[e5,t]

Output =

               2                              2
X''[t] == -(r w  Cos[t w]) - L Cos[f[t]] f'[t]  - 
 
   L Sin[f[t]] f''[t]