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

step 1 In this question it is given a rope under the tension of 200 Newton so t is equal to 200 Newton

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

A rope, under 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 $y=(0.10 \mathrm{~m}) \sin \left(\frac{\pi x}{2}\right) \sin (12 \pi t)$, where $x=0$ at one end of the rope, $x$ is in metres, and $t$ is in seconds. Find (A) the length of the rope (B) the speed of waves on the rope (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 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 $y=(0.10 \mathrm{~m}) \sin \left(\frac{\pi x}{2}\right) \sin (12 \pi t)$, where $x=0$ at one
end of the rope, $x$ is in metres, and $t$ is in seconds. Find
(A) the length of the rope
(B) the speed of waves on the rope
(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 patterns that result from the interference of two waves traveling in opposite directions along a medium. In systems with fixed endpoints, such as a rope fixed at both ends, the standing wave exhibits nodes (points of zero amplitude) at the boundaries and antinodes where the amplitude is maximum, with the overall pattern determined by the characteristics of the wave on that medium.

Harmonics in Fixed Strings

Harmonics refer to the specific modes of vibration that occur at discrete frequencies in a system with fixed boundaries. For a string fixed at both ends, the nth harmonic is characterized by n half-wavelengths fitting along the string’s length. This relationship is essential for connecting the spatial description of the wave (through its wavelength) with the physical length of the medium and the mode of oscillation.

Wave Equation and Wave Speed

The wave equation describes how disturbances propagate through a medium and relates the wave’s frequency, wavelength, and speed through the equation v = f?. In the context of standing waves, the spatial part of the solution identifies the wavelength and nodes, while the time-dependent part identifies the oscillation frequency. This relationship is fundamental in analyzing any vibrating system.

Tension and Mass per Unit Length

The speed of waves on a string or rope under tension is determined by both the tension applied and the mass per unit length of the medium. This relationship is given by v = ?(T/?), where T is the tension and ? is the linear mass density. This formula is crucial for linking measurable physical properties such as tension and mass to the observed wave speed.

Frequency, Wavelength, and Period Relationships

Frequency, wavelength, and period are interrelated characteristics of wave motion. The frequency (f) is the number of oscillations per second, the wavelength (?) is the distance over which the wave repeats, and the period (T) is the time for one complete oscillation (with T = 1/f). Understanding these relationships is important for analyzing how changes in one property (such as harmonic number) affect the others in a standing wave system.

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