Question

Derive a formula for the maximum speed vmax of a simple pendulum bob in terms of g, the length ℓ, and the maximum angle of swing θmax. NOTE!! When entering a trig function of an angle, use parentheses around the argument of the trig function. For instance, enter cos( θmax) rather than cos θmax.

Answer 1

when the pendulum bob is at rightmost position making angle θ_max with the vertical

in triangle OBC

Cosθ_max = OC/OB

Cosθ_max = OC/L

OC = L Cosθ_max

height gained relative to the lowest position is given as

h = OA - OC = L - L cos$\theta$

Using conservation of energy

Potential energy = kinetic energy

(0.5) m V² = mgh

V² = = 2g(L - L cos$\theta$)

V_max = sqrt (2g(L - L cos$\theta$))