A pendulum consists of a ball at the end of a massless string of length
1.4 m. The ball is released from rest with the string making an angle of 20
degrees with the vertical. What is the maximum speed of the pendulum?

Use D0EL (energy conservation).

Step 1: Define/draw system and coordinates. (see transparency)

Initial position i and final position f (at the bottom).

Step 2: Define potential energy zero.

Define the zero of the potential energy to be the bottom of the path.

Step 3: Energy conservation statement.

**U _{i} + K_{i} = U_{f} +
K_{f} **

Initial kinetic energy K_{i} = 0 because the ball is released from rest.

Final potential energy U_{f} = 0 by definition.

Conclusion: U_{i} = K_{f}
or
mgh = ½ mv_{f}^{2},

where h is the height from which the ball is released.

Cancel the m's to give: v_{f} =
(2gh)^{½ }

Determine h from geometry: h = L - Lcos θ = L (1-cos θ)

With L = 1.40 m and θ = 20^{o}, h = 0.0844 m,
v_{f} = (2gh)^{½ }
= 1.29 m/s