True: But the complete solution to this problem is
1) Use ODE
yielding Theta(t)= Theta0*cos(w*t) w=Sqrt(g/L):
From ODE: d2(Theta)/dt2+(g/L) *(Theat)=0: Homogeneous Response, Initial Conditions yield Theta(t) above
2) Apply Dynamic Equations in 2 D Polar: these ACCELERATIONS[relative to the Rotating COORDINATE SYSTEM] are:
Remember that "w" is from solution of the ODE and is a function of time : w(t)=d(Theta(t))/dt
Radially= (dL2/d2t)-(wsquared)*L :Radial Acceleration/Centripital Acceleration
Tangentially=[d2(Theta)/dt2]L+2(Theta/dt)*L: Tangential Acceleration/Coriolis Acceleration.
In the case of a Pendulum, Radial and Coriolis are zero.
Now just "plug and chug" w(t) into the Dynamic Equations for Tangential and Centripital Acceleration.
NOTE: This can easily be done from Velocity , as well ,since:
And successive differentiation BUT REMEMBER YOU MUST DIFFERENTIATE THE RADIAL AND TANGENTIAL UNIT VECTORS DUE TO ROTATION:
d(Tangentialunitvector)/dt=-w*Radialunitvector Again: (d(Theta(t))/dt=w(t)
Hope this helps:
PPM Ph.D Applied Physics