9.) A weight attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidally with time. You start a stopwatch. When the stopwatch reads 0.3 seconds, the weight first reaches a high point 60 cm above the floor. The next low point, 40 cm above the floor, occurs at 1.8 seconds. Write an equation that models the distance from the floor with respect to time.

To model the distance from the floor, we first need to determine the parameters of a sinusoidal function. The average height (midline) can be calculated as the midpoint between the high point (60 cm) and the low point (40 cm), which is 50 cm. The amplitude, the distance from the midline to either the high or low point, is 10 cm (60 cm - 50 cm or 50 cm - 40 cm). Next, we find the period, which is the total time it takes for one complete cycle. From the first high (0.3 seconds) to the low (1.8 seconds) is 1.5 seconds, suggesting that a full cycle (high to low to high) takes about 3 seconds. The frequency is thus \( \frac{2\pi}{3} \). Finally, we incorporate the phase shift. Since the first high occurs at 0.3 seconds, we can write the equation as follows: \[ d(t) = 50 + 10 \cos\left(\frac{2\pi}{3}(t - 0.3)\right) \] This function models the distance from the floor with respect to time, capturing the sinusoidal motion of the bouncy weight! Want to put that bouncing weight to practical use? You could create a simple energy conversion demonstration! Attach a small weight, an LED light, or even a small wind-up toy to the spring. As the weight bounces and compresses the spring, it not only helps visualize energy transfer but could also serve as a quirky desk toy. Plus, if you time its oscillations, you can impress friends with your newfound knowledge of physics in action! Got a budding interest in oscillations and waves? There’s an array of fascinating resources out there. Check out classic physics texts, like “The Feynman Lectures on Physics,” for a deeper dive into wave phenomena, or explore online platforms like Khan Academy, which offer interactive lessons on harmonic motion. Also, a book like “The Physics of Vibrations and Waves” by H. J. Pain can provide insight into how waves behave in various scenarios!