Draw a free-body diagram and write Newton’s second law of motion for the forces...

Question

Draw a free-body diagram and write Newton’s second law of motion for the forces and accelerations experienced by a pendulum bob of mass m swinging under the influence of gravity, g, when it is at the very bottom of its swing (θ=0°). There are two force terms that need to be considered (be careful about the direction of those forces) and only one acceleration term. Consult your unit on circular motion for this. Finally, using this equation, derive an expression for the velocity of the bob at that point.

Answer 1

In general, the 2D force vectors are the string tension T = (–sinθ,cosθ)T and the weight W = (0,–mg), and the acceleration vector is centripetal,

a = d²r/dt² = dv/dt = ((cosθ,sinθ)d²θ/dt² + (–sinθ,cosθ)(dθ/dt)²)L,

where

r = (sinθ,1 – cosθ)L

is the 2D location vector, L is the string length, and

v = dr/dt = (cosθ,sinθ)Ldθ/dt

is the 2D velocity vector.