For the following problem use \( T=2 \pi \sqrt{\frac{L}{g}} \). Let \( \pi \approx 3.14 \) and \( g=32 \) feet per second \( { }^{2} \). A child is swinging on a rope 96 feet long over a river swimming hole. How long does it take (in seconds) to complete one swing back and forth?

To find the period \( T \) of the swing, we use the formula \( T = 2 \pi \sqrt{\frac{L}{g}} \). Here, \( L = 96 \) feet and \( g = 32 \) feet per second squared. First, let's calculate \( \frac{L}{g} \): \[ \frac{L}{g} = \frac{96}{32} = 3. \] Now, we take the square root: \[ \sqrt{\frac{L}{g}} = \sqrt{3} \approx 1.73. \] Then, we multiply this by \( 2 \pi \): \[ T \approx 2 \times 3.14 \times 1.73 \approx 10.86 \text{ seconds.} \] So, it takes approximately 10.86 seconds to complete one swing back and forth! You’re essentially riding a joyous pendulum, swinging across time with each whoosh over the river. Isn't physics a blast? And next time you swing, you can impress your friends with a mind-blowing calculation! Now, think of this: if a child is on a swing for only one complete motion, they might not even realize they're experiencing a whole mini-lifetime on that side-to-side journey. Swinging is not just play; it’s a perfect blend of science and fun!