In a demonstration, a 1.2 kg horizontal rope is fixed in place at its two ends (x=0 and x=2.0 m)

In a demonstration, a 1.2 kg horizontal rope is fixed in...

In a demonstration, a 1.2 $\mathrm{kg}$ horizontal rope is fixed in place at its two ends $(x=0$ and $x=2.0 \mathrm{m})$ and made to oscillate up and down in the fundamental mode, at frequency 5.0 $\mathrm{Hz}$ . At $t=0,$ the point at $x=1.0 \mathrm{m}$ has zero displacement and is moving upward in the positive direction of a $y$ axis with a transverse velocity of 5.0 $\mathrm{m} / \mathrm{s}$ . What are (a) the amplitude of the motion of that point and (b) the tension in the rope? (c) Write the standing wave equation for the fundamental mode.

In a demonstration, a 1.2 $\mathrm{kg}$ horizontal rope is fixed in
place at its two ends $(x=0$ and $x=2.0 \mathrm{m})$ and made to oscillate up and down in the fundamental mode, at frequency 5.0 $\mathrm{Hz}$ .
At $t=0,$ the point at $x=1.0 \mathrm{m}$ has zero displacement and is moving upward in the positive direction of a $y$ axis with a transverse velocity of 5.0 $\mathrm{m} / \mathrm{s}$ . What are (a) the amplitude of the motion of that point and (b) the tension in the rope? (c) Write the
standing wave equation for the fundamental mode.

Key Concepts

Standing Waves

Standing waves form when two waves of the same frequency and amplitude travel through a medium in opposite directions and interfere with each other. In a rope fixed at both ends, the reflections create a pattern of nodes (points with no displacement) and antinodes (points of maximum displacement). This concept is fundamental to analyzing oscillations in constrained systems as it dictates how energy is stored and distributed.

Fundamental Mode

The fundamental mode, or first harmonic, is the simplest vibration pattern of a string fixed at both ends. It features nodes at both ends and a single antinode in the middle. This mode corresponds to the lowest possible frequency at which the system can oscillate, and its characteristics directly influence calculations of quantities like amplitude and tension.

Wave Equation

The wave equation describes how wave phenomena propagate through a medium. In the case of a vibrating rope, it governs the spatial and temporal variation of displacement. For standing waves, the solution takes the form of sinusoidal functions in both space and time, and it is essential for deriving expressions that model the observed oscillating behavior under boundary conditions.

Tension and Mass Density Relationship

The speed of a wave in a rope is determined by the tension in the rope and its linear mass density. This relationship, often expressed as v = ?(T/?), where T is the tension and ? is the mass per unit length, is critical in linking the mechanical properties of the system to the dynamics of wave propagation. It allows one to compute the required tension for a given wave speed or frequency.

Transverse Motion Characteristics

In harmonic motion, the maximum transverse velocity of a point on the rope is linked to both the amplitude of motion and the angular frequency. When the displacement is zero at a node, the velocity reaches its maximum value. This relationship is used to determine the amplitude from the given maximum velocity and connects the kinematics of the oscillation with its spatial profile.

Transcript

00:01 So in this question, we have this rope that is 1 .2 kilogram and it's fixed between two ends, that's 2 meters apart, and it has a frequency of 5 hertz when it's oscillating in the fundamental mode.

00:13 So fundamental mode looks something like this.

00:17 So in a fundamental mode, we know the wavelength is twice the length of the length of this rope.

00:24 So we say that at t equals 0, the rope at 1 meter, the point at x equals 1 meter, has zero displacement.

00:38 And then this rope is moving upward in the positive direction y -axis, and it has a transverse velocity of 5 meter per second.

00:54 So from this information, we want to find what is the amplitude of the motion of that point? amplitude ym.

01:04 Part b, what is the tension in the rope? part c, what is the equation for the standing wave? okay, so let's first look at this part.

01:17 We can write out the equation that we usually have for a standing wave.

01:23 So the standing wave has this equation of y equals amplitude, sine kx, sine omega -t.

01:37 And then it also has a speed, u of u is, you take the derivative of y and you get, omega times y m, sine kx, cosine omega t.

01:58 So it also has this maximum speed of omega times y n, when the sine cosine part has 1.

02:08 So the wave speed, v, is tau over mu, square root, tau over mu.

02:24 Alright, so i think these are all the information we will need for this question.

02:30 I will put everything in this corner so that we can refer back to it...

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