A pendulum swings back and forth once per second. The pendulum is shortened by removing half of the string. How often will the pendulum swing back and forth in a minute?

To determine how often the pendulum swings back and forth after being shortened, it's essential to understand the relationship between the length of a pendulum and its period (the time it takes to complete one full swing).

The period of a simple pendulum is given by the formula:

[ T = 2\pi \sqrt{\frac{L}{g}} ]

where ( T ) is the period, ( L ) is the length of the string, and ( g ) is the acceleration due to gravity. When the length of the pendulum is halved, the new period becomes:

[ T' = 2\pi \sqrt{\frac{L/2}{g}} = 2\pi \sqrt{\frac{L}{2g}} = \frac{T}{\sqrt{2}} ]

This indicates that the period decreases, and the pendulum swings back and forth more frequently. Specifically, if the original pendulum swung once per second, the new period (after halving the length) is approximately 0.707 seconds (since ( \sqrt{2} \approx 1.414)).

The frequency in swings per second is the inverse of the period. Therefore, if the