(Photo of the word problem) ~ h t t p s : //imgur. c o m/a/cZ0htfE
The figure above on the left shows a uniformly thick rope hanging vertically from an oscillator that is turned off. When the oscillator is on and set at a certain frequency, the rope forms the standing wave shown above on the right. P and Q are two points on the rope.
a) The tension at point P is greater than the tension at point Q. Briefly explain why.
b) A student hypothesizes that increasing the tension in a rope increases the speed at which waves travel along the rope. In a clear coherent paragraph-length response that may also contain figures and/or equations, explain why the standing wave shown above supports the student's hypothesis.
a) The tension at point P is greater than the tension at point Q because point P is a point of maximum displacement in the standing wave, while point Q is a point of zero displacement. In a standing wave, the rope oscillates between points of maximum displacement (called antinodes) and points of zero displacement (called nodes). At the antinode, the rope experiences maximum displacement, and therefore, a greater tension force is exerted on the rope due to the increased stretching and compression. On the other hand, at the node, the rope does not move at all, resulting in a lower tension force.
b) The standing wave shown above can support the student's hypothesis that increasing the tension in a rope increases the speed at which waves travel along the rope. The wave speed in a medium is determined by the tension in the medium and the density of the medium. According to the equation v = √(T/μ), where v represents the wave speed, T represents the tension, and μ represents the linear mass density of the rope, the wave speed is directly proportional to the square root of the tension in the rope. As the tension increases, so does the wave speed.
In the given standing wave, we observe that the amplitude (maximum displacement) of the wave increases as we move from the node (point Q) towards the antinode (point P). This suggests that the tension in the rope is higher at point P compared to point Q. Since the wave speed is directly related to the tension, the observed increase in amplitude supports the student's hypothesis that increasing the tension in a rope would increase the speed at which waves travel along the rope. This is because the greater tension causes the wave to propagate faster through the medium.