Bob Goes Faster
Scenario A: Bob attaches a mass \(m\) at the end of a long string (length \(L\)) and holds the other end of the string, allowing the mass to swing like a pendulum, with amplitude \(A\). When the mass reaches a turning point, Bob quickly pulls on the string so that the swinging part of the string shortens to \(\tfrac12 L\). The pendulum now swings with a higher frequency.
Scenario B: Bob repeats the exact same situation: a pendulum length \(L\) and mass \(m\) is swinging with amplitude \(A\). This time, Bob quickly pulls on the string when the mass passes through the equilibrium position. Again, the string is shortened to \(\tfrac12 L\), and the pendulum swings with a higher frequency.
In which scenario, if any, will the pendulum have a greater amplitude after Bob shortens the string?
Assumptions: The size of the swinging mass is negligible. When Bob pulls on the string, he does this in a negligible amount of time. During the pull, the force is directed upward along the string.