Second Harmonic A rope, under a tension of 200 N and fixed at both ends, oscillates in a second

step 1 In the party of this problem, we have to find the length of the rope. So let's start by writing

Second Harmonic A rope, under a tension of 200 N and fixed...

Second Harmonic 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 $$ y(x, t)=(0.10 \mathrm{~m})\left(\sin \left(\left(\frac{\pi}{2} \mathrm{rad} / \mathrm{m}\right) x\right) \sin ((12 \pi \mathrm{rad} / \mathrm{s}) t)\right. $$ where $x=0.00 \mathrm{~m}$ 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?

Second Harmonic 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
$$
y(x, t)=(0.10 \mathrm{~m})\left(\sin \left(\left(\frac{\pi}{2} \mathrm{rad} / \mathrm{m}\right) x\right) \sin ((12 \pi \mathrm{rad} / \mathrm{s}) t)\right.
$$
where $x=0.00 \mathrm{~m}$ 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

Standing Waves

Standing waves are formed by the interference of two waves traveling in opposite directions, resulting in wave patterns with fixed nodes and moving antinodes. They occur under specific boundary conditions and are key to understanding resonance phenomena in physical systems.

Harmonic Standing Wave Patterns

Harmonic standing wave patterns refer to the specific configurations in which an integral number of half-wavelengths fit into a bounded medium, such as a rope with fixed ends. The fundamental mode is the first harmonic, and higher harmonics (overtones) show more nodes and antinodes, with each successive harmonic adding an extra half-wavelength.

Wave Speed on a Rope

The wave speed on a rope is determined by its tension and its linear mass density, following the relationship v = ?(T/?). This concept connects the mechanical properties of the rope to the dynamics of wave propagation along it.

Linear Mass Density

Linear mass density is the mass per unit length of a rope or string. It is a fundamental parameter in the study of waves on strings because it directly affects the wave speed and is crucial in determining how much mass participates in the oscillation.

Wavelength, Frequency, and Period Relationships

The interrelation between wavelength, frequency, and wave speed is central to understanding wave behavior. Specifically, the equation v = f? links these quantities, while the period of oscillation is the reciprocal of the frequency. These relationships enable the calculation of other wave parameters when one or more are known.

Transcript

00:03 In the party of this problem, we have to find the length of the rope.

00:12 So let's start by writing the strening equation for the stranding waif as y of x and t is equals to 0 .10 meter into sine of pi by 2 radian per meter into x into sine of omega, which is equals to 12 pi radian per second, to t we call it equation number one now the general form of this standing wave equation is written as y of x n t is equals to ym sine of kx and sine of omega t so this will be our equation number two in this case this ym is the amplitude k is the web number and this omega is the angular frequency of the web by comparing these two equal we can write the value for this k as k is equal to the definition of k is that it is a wave number which is equals to 2 pi divided by lambda so from here we can write this lambda as lambda is equals to 2 pi divided by k and from the above two equations we can write this one is 2 pi divide by k which is equals to pi divide by 2 so from here we will get the value for this lambda is 4 .0 meter.

02:02 So this is the value for the waft length.

02:07 Now for second harmonic, for second harmonic we can write the length as length is equal to lambda 2 and this is equals to 4 .0 meter.

02:22 So the length of the rope is equals to 4 .0 meter.

02:27 Let's move to the part b of this problem.

02:31 In this case we are going to find the speed of the waft on the rope.

02:35 Let the speed of the waif on the rope is v.

02:39 We can write the speed of the waif as v as v as v.

02:42 V.

02:42 2.

02:43 And from the above two equations, we can write the value for this omega, which is equals to 2 pi f2.

02:52 So this is equals to 12 pi radiant per second.

02:57 So from here, we will get the value for this f2 as f2 is equals to 6 .0 hertz.