A rope, under a tension of 200 N and fixed at both ends, oscillates in a second-harmonic standin

A rope, under a tension of 200 N and fixed at both ends,...

A rope, under a tension of 200 $\mathrm{N}$ and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by $$y=(0.10 \mathrm{m})(\sin \pi x / 2) \sin 12 \pi t$ where $x=0$ at one end of the rope, $x$ is in meters, and $t$ is in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

A rope, under a tension of 200 $\mathrm{N}$ and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by $$y=(0.10 \mathrm{m})(\sin \pi x / 2) \sin 12 \pi t$ where $x=0$ at one end of the rope, $x$ is in meters, and $t$ is in seconds. What are (a) the length of the rope, (b) the speed of the
waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the
period of oscillation?

Key Concepts

Linear Mass Density

Linear mass density is the mass per unit length of a rope or string. It plays a crucial role in wave dynamics, as it directly affects the wave speed and is essential when relating measured properties of wave motion to the actual mass of the physical medium.

Wave Speed on a Tensioned String

The speed of a wave traveling along a rope or string under tension is determined by the relationship v = ?(T/?), where T is the tension and ? is the linear mass density. This expression links the physical properties of the rope to how quickly disturbances propagate along it.

Wavelength-Frequency Relationship

For any periodic wave, the relationship v = f? (where v is the wave speed, f is the frequency, and ? is the wavelength) is fundamental. This relation allows for determining one of these quantities if the other two are known, and is particularly useful in analyzing harmonic patterns in systems with fixed lengths.

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere. In systems with fixed boundaries, this interference results in a pattern of nodes (points of no motion) and antinodes (points of maximum motion). This concept is fundamental in understanding the discrete vibration patterns seen in systems like ropes or strings.

Harmonics

Harmonics refer to the distinct natural modes of vibration of a system, each corresponding to a specific frequency. The fundamental (first harmonic) is the simplest mode of vibration, and higher harmonics (second, third, etc.) have increasingly complex patterns with more nodes and antinodes. The spacing and number of these nodes are determined by the boundary conditions of the system.

Transcript

0:00 16 .52.

00:01 So we've got a rope that's under a tension of 200 newtons and it's fixed at both ends.

00:06 We're told that it's oscillating in its second harmonic pattern for the standing wave.

00:12 Now we're given the function for the displacement and we'd like to find out some things about this rope.

00:19 Like we want to find its length, the wave speed, the mass of the rope, and then what the period is of the third harmonic standing wave pattern.

00:33 So first, for the second harmonic, we know that we're going to fit two half -wave lengths in.

00:48 Because for a string that's fixed at both ends, the number of the harmonic is the number of half -wave lengths.

00:57 So the wavelength is just going to be the overall length of the rope.

01:02 The frequency, as we know, is the speed divided by the length.

01:08 It's the speed divided by the wavelength, but in this case it's just the length of the rope.

01:12 So here we can see that the wave number is pi over two radians per meter.

01:29 We know the wave number is also equal to 2 pi divided by the wavelength, and we know that the wavelength is the length of the rope.

01:44 So then just rearranging this for l, we find that the length of the rope is 4 meters.

02:00 Now we know omega from this is 12 pi radians per second, which is also 2 pi -f.

02:16 We could in fact just divide the, you know what, actually let's just think of it that way...

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