AP Physics 1 2015 Free Response Question 5

Question: 
A string is attached to an oscillator and its other end is attached to a block as shown below. The string passes over a pulley that is massless and frictionless. The distance between the oscillator and pulley is L and the mass of the block is M. The driving frequency of the oscillator is adjusted such that the string vibrates in its second harmonic. Identify locations on the string that have the greatest vertical speed.

Scoring Guidelines:

An indication that the string vibrates in its second harmonic or a wave is drawn such that λ2 = L.	1 point
Indicate points that are at the antinodes of any standing wave drawn on the string. (Full credit: the two points are located at one-fourth length of the string and three-fourths length of the string from the oscillator.)	1 point

(Source: http://apcentral.collegeboard.com/home)

Feynman's insights or goofs?:
The purpose of this question is to examine the properties of standing waves and apply the appropriate relationships among wavelength, frequency, and wave speed. Furthermore, students are expected to locate the nodes and antinodes of a string that is oscillating.

In this experiment, a string with one end is attached to an oscillator and its other end is attached to a block. Importantly, a node is not exactly located at the end that is attached to the oscillator because the oscillator vibrates up and down continuously. A definition of nodes is “[t]he points where there is no motion (Feynman et al., 1963, section 49-1 The reflection of waves).” If we observe the oscillations more carefully, the (virtual) node could be located slightly to the left of the oscillator. In a similar sense, an antinode does not occur exactly at the opening of the open pipe, but slightly beyond the opening where there is a lesser constraint for the air molecules to oscillate (See figure 1 below). The extra distance needed is sometimes called an end correction for the open-ended pipe. Thus, we should include the end correction for a more accurate calculation of the resonance frequency of the pipe. 

Fig 1

The scoring guidelines state that the two antinodes are located at one-fourth length of the string and three-fourths length of the string from the oscillator. Strictly speaking, the node does not occur exactly at the oscillator, and the antinodes can be located slightly to left of the one-fourth length of the string and three-fourths length of the string from the oscillator (See figure 2 below). Essentially, the points on the string that have the greatest average vertical speed can be located slightly to the left of the expected antinodes because of the end effect. Therefore, students could be penalized if the anti-nodes are not drawn exactly at the expected locations. Furthermore, some students might not indicate a node occurs at the oscillator which is not incorrect. 

Fig 2

In Feynman’s own words, “[s]uppose that the string is held at one end, for example by fastening it to an ‘infinitely solid’ wall. This can be expressed mathematically by saying that the displacement y of the string at the position x = 0 must be zero because the end does not move (Feynman et al., 1963, section 49–1 The reflection of waves).” However, there is a continuous vertical movement for the segment of the vibrating string attached to the oscillator. Thus, the node does not strictly occur at the location of the oscillator because the displacement of this segment of the string is not always zero. Furthermore, Feynman adds that “[t]he points where there is no motion satisfy the condition sin (ωx/c) = 0, which means that (ωx/c) = 0, π, 2π, …, nπ, … These points are called nodes (Feynman et al., 1963, section 49–1 The reflection of waves).” In short, the nodes should be defined as points that are always stationary.

Interestingly, Feynman mentions that “if we assume that the string is infinite and that whenever we have a wave going one way we have another one going the other way with the stated symmetry, the displacement at x = 0 will always be zero and it would make no difference if we clamped the string there (Feynman et al., 1963, section 49–1 The reflection of waves).” In a sense, Feynman has a slip of tongue when he simply says that “the string is infinite.” To be precise, the vibrating string is not infinitely long or massive, but that it is attached to an “infinitely rigid” wall. Thus, physics teachers could explain that Feynman is sometimes sloppy in his language usage.

More importantly, the fundamental frequency of the vibrating string is not simply dependent on the linear mass density and tension in the real world. Feynman explains that “[t]he idea that the natural frequencies are harmonically related is not generally true. It is not true for a system of more than one dimension, nor is it true for one-dimensional systems which are more complicated than a string with uniform density and tension. A simple example of the latter is a hanging chain in which the tension is higher at the top than at the bottom (Feynman et al., 1963, section 49–3 Modes in two dimensions).” Nevertheless, the stiffness of a real-life string can cause the wave velocity to be also dependent on the wavelength (Elmore & Heald, 1985).

Note: 

1. A node can be distinguished as a displacement node or a pressure node. For example, a “pressure node (corresponding to a displacement or velocity antinode) occurs at the open end of a tube, while a pressure antinode (corresponding to a displacement or velocity node) occurs at the closed end (Gregersen, 2011, p. 37).”

2. For another discussion of this question, you can visit the following website:
https://www.youtube.com/watch?v=z_KX8Xpxa-c

References:

1. Elmore, W. C., & Heald, M.A. (1985). Physics of Waves. New York: Dover.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
3. Gregersen, E. (2011). The Britannica Guide to Sound and Light. New York: The Rosen Publishing Group.