You must give a unit in all of your numeric answers in order to obtain the mark. Part A Distance from nodes to antinodes Two waves of amplitude 4.0cm and frequency 14 Hz are moving in opposite directions at 5.6ms^-1 along a stretched string If a standing wave were formed, how far apart would you expect the antinodes to be from the nodes on either side of them? Value square Units Hint1 Hint 2 If the string had two fixed ends, what is the minimum length it must be in order for a standing wave to be possible? square Hint 1 Hint 2

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You must give a unit in all of your numeric answers in order to obtain the mark. Part A Distance from nodes to antinodes Two waves of amplitude 4.0cm and frequency 14 Hz are moving in opposite directions at 5.6ms^-1 along a stretched string If a standing wave were formed, how far apart would you expect the antinodes to be from the nodes on either side of them? Value square Units Hint1 Hint 2 If the string had two fixed ends, what is the minimum length it must be in order for a standing wave to be possible? square Hint 1 Hint 2

Answer 1

For Part A - Distance from nodes to antinodes First, in general, we need to establish that the wavelength (\(\lambda \)) of a wave case as follows can be calculated using the formula \(\lambda = \text{speed/frequency}\). Once we have the wavelength, understanding that the distance from every node to its adjacent antinode is half of this wavelength will enable us solve this problem. For this question, since we know the speed of the wave (5.6 m/s) and frequency of the wave (14 Hz), we can calculate the wavelength as follows: \(\lambda \) = speed/frequency = 5.6 m/s / 14 Hz = 0.4 m The distance between a node and its adjacent antinode is half of the wavelength (\( \lambda/2 \)), since a complete cycle (peak to zero to negative peak to zero) is a full wavelength, and (peak to zero) or (zero to peak) would make it just half of the full cycle. Therefore: Distance(Node to Antinode) = \( \lambda/2 \) = 0.4m / 2 = 0.2m For Part B - Minimum length of the string The minimal requirement to establish a standing wave pattern on any stretched string is one full length of wave on that string, which equivalently stands as twice the distance between node and antinode. If the string many meet up this minimum length, no practical standing wave pattern could be established. Therefore, because we've already noted that half wavelength is the distance between node and antinode (0.2m), minimum string length, or the minimum path distance required to hold one full cycle of the wave (wavelength), is calculated as twice that value, i.e: Minimum String Length = \( 2 \times \) distance (Node to Antinode) = \( 2 \times 0.2m \) = 0.4m So the solutions to your questions are: 【Part A - Answer】: the distance from nodes to their adjacent antinodes in a standing wave - 0.2 \(\text{m}\) 【Part B - Answer】: minimum length string where a standing wave can be obtained - 0.4 \(\text{m}\)

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