Two children are having a splashing competition using a diving board. The first child is very small and jumps lightly off the end of the diving board. The second child, who is much larger, jumps heavily off the board.
A. In what way or ways, if any, are the vibrations of the board the same in the two cases? Explain your reasoning.
B. In what way or ways, if any, are the vibrations of the board different in the two cases? Explain your reasoning.
A. The vibrations of the board in both cases are likely to be similar in frequency. When the first child jumps lightly off the end of the diving board, the force applied to the board may be smaller, but the frequency at which the board vibrates should be similar. The frequency of the vibrations depends on the physical properties of the board and how it responds to external forces, such as a person jumping on it. Assuming the board is made of the same material and has similar dimensions, the frequency of the vibrations should be similar for both children.
B. The vibrations of the board in the two cases can differ in their amplitude, or the extent of displacement of the board from its resting position. When the second child, who is larger, jumps heavily off the board, the force applied to the board is greater. This increased force can lead to larger amplitude vibrations, resulting in greater displacement of the board from its resting position compared to when the smaller child jumps lightly. This difference in amplitude can be observed visually or by measuring the displacement of the board at different points. However, the frequency of the vibrations, as mentioned earlier, is likely to be similar for both cases.
## Similarities in Vibrations:
* **Frequency:** Both children will cause the board to vibrate at **its natural frequency**, regardless of their weight. This is because the board has a specific resonant frequency determined by its material, length, and stiffness. So, the up-and-down oscillations of the board will occur at the same rate for both jumps.
* **Direction:** The vibrations will oscillate in the **same direction** (typically up and down) for both jumps. The board will bend downwards at the tip when the child pushes off, then swing back up, and repeat this cycle until the vibrations gradually dissipate.
## Differences in Vibrations:
* **Amplitude:** The board's **vibrations will be much larger** for the heavier child. This is because the force applied by the larger child is greater, causing the board to bend further and create larger amplitude oscillations. This means the up-and-down movement of the tip will be significantly higher in the second case.
* **Duration:** The vibrations will also **last longer** after the heavier child jumps. This is because more energy is transferred to the board, meaning it takes longer for the vibrations to die down due to internal friction and air resistance.
* **Higher harmonics:** While the dominant vibrations will be at the board's natural frequency, the heavier child's jump might also excite **higher harmonic frequencies**. These are additional, smaller vibration patterns that occur simultaneously with the main oscillation. This is because the larger force is more likely to introduce non-linear effects in the board's bending, creating complex vibration patterns.
Therefore, while the vibrations share some similarities in terms of frequency and direction, the force applied by the heavier child significantly impacts their amplitude, duration, and complexity.