Solution

Well what forces are acting on the pendulum? We're ignoring friction, so the only forces are the tension and the force of gravity. The tension acts perpendicular to the direction of motion and hence does no work. The force of gravity is conservative, so we can use conservation of energy to solve this problem.

Define $y=0$ at the point the pendulum is attached to the ceiling. We'll first talk about the initial state of the pendulum, and then its final state.

initial time:

The initial y value of the pendulum is $y_0 ~=~ -L\cos\theta$. Therefore the initial potential energy is $-mgL\cos\theta_0$. The initial kinetic energy is zero. Therefore the total energy is $K+U ~=~ -mgL\cos\theta_0$.

final time:

The final value of y is $-L$ and we'll call its final kinetic energy ${1\over 2}mv^2$, so the final energy is ${1\over 2}mv^2 + (-mgL)$

The initial and final energies are the same, by conservation of mechanical energy. Therefore we can solve for $v$:

$$v ~=~ \sqrt{2gL(1-\cos\theta_0)}$$ (1.19)