91Ó°ÊÓ

Suppose that a point \(a\) on a string at \(z_{n}=10 \mathrm{~cm}\) oscillates in harmonic motion at frequency 10 eps with amplitude \(1 \mathrm{~cm}\). Its phase is such that at \(t=0\) the point on the string is passing through its equilibrium position with upward velocity (positive displacement is upward). (a) What is the magnitude and direction of the velocity of point \(a\) at \(t=0.05 \mathrm{sec}\) ? Suppose the string parameters (mass per unit length and tension) are such that the wave velocity is \(100 \mathrm{~cm} / \mathrm{sec}\) (b) What is the wavelength of a traveling wave? What is the wavelength of a standing wave? (c) Another point \(b\) at \(z_{b}=15\) cm oscillates with the same amplitude as that at \(z_{a}=10 \mathrm{~cm}\), but with a relative phase of 180 deg with respect to the oscillation at \(\hat{ }_{20}\) Can you tell whether we have here a pure traveling wave, a pure standing wave, or a combination? (d) A third point \(c\) at \(12.5 \mathrm{~cm}\) also oscillates with the same amplitude as that at \(z_{n}\) but 180 deg out of phase with point \(a\). Point \(b\) oscillates as given above. Now tell us whether the wave is a traveling or a standing wave (or a combination).

Step by step solution

01

Understand Harmonic Motion

The given point \(a\) oscillates in harmonic motion with amplitude \(1\, \text{cm}\), frequency \(10\, \text{eps}\), and phase implying it starts at equilibrium with positive velocity. The displacement \(y(t)\) can be described by \(y(t) = A \sin(\omega t + \phi)\) where \(A = 1\, \text{cm}\), \(\omega = 2\pi \times 10\, \text{rad/s}\), and initial phase \(\phi\) is 0.

02

Calculate Angular Frequency (\(\omega\))

Given frequency \(f = 10\, \text{eps}\), we calculate the angular frequency \(\omega = 2 \pi f = 20\pi\, \text{rad/s}\).

03

Determine Velocity Equation

The velocity \(v(t)\) is the first derivative of \(y(t)\): \(v(t) = \frac{dy}{dt} = A \omega \cos(\omega t + \phi) = 1 \cdot 20\pi \cos(20\pi t)\).

04

Evaluate Velocity at \(t = 0.05\,s\)

Plug in \(t = 0.05\, \text{s}\) into \(v(t) = 20\pi \cos(20\pi \times 0.05)\). Simplifying, \(v(0.05) = 20\pi \cos(\pi) = -20\pi\). Since \(\cos(\pi) = -1\), \(v(0.05) = -20\pi\, \text{cm/s}\) indicating downward velocity.

05

Calculate Wavelength for Traveling Wave

For a speed \(v = 100\, \text{cm/s}\), and frequency \(f = 10\, \text{eps}\), use \(\lambda = \frac{v}{f} = \frac{100}{10} = 10\, \text{cm}\).

06

Analyze Standing vs. Traveling Wave for Point \(b\)

At point \(b\), the oscillations have a 180-degree phase difference. This phase shift and same amplitude between points indicate the possibility of a standing wave pattern.

07

Analyze Points \(a\), \(b\), and \(c\) for Wave Type

At point \(c\), 180 degrees out of phase with \(a\), and \(b\) further complicates the phase pattern. The relative phasing, amplitude, and pattern suggest a combination wave rather than purely traveling or standing.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Harmonic Motion

Harmonic motion is a type of periodic motion where an object moves back and forth in rhythmic cycles. This movement can be described by sinusoidal functions, often appearing as a sine or cosine wave. Consider a point on a string oscillating in harmonic motion. This means the point moves to and fro around an equilibrium position, just like a pendulum.

• The amplitude is the maximum distance from the equilibrium position. In this exercise, it's 1 cm.
• Frequency tells us how often the cycle repeats in one second. Here, 10 cycles per second or 10 eps.
• Initial phase is crucial as it determines the starting position of the motion. The point initially starts at equilibrium with a positive velocity.

For our exercise, the motion can be described with the equation: \[ y(t) = A \sin(\omega t + \phi) \] where \(\omega\) is the angular frequency, and \(\phi\) is the phase. The angular frequency \(\omega = 2\pi f\) shows how fast the point oscillates.

Wave Velocity

Wave velocity refers to how fast a wave propagates through a medium. It is crucial for understanding how energy is transmitted across the wave, such as on a string. The formula to calculate wave velocity is: \[ v = \frac{\lambda}{T} = \lambda f \]where:

• \(v\) is the wave velocity.
• \(\lambda\) is the wavelength.
• \(T\) is the period of the wave—the time it takes for one cycle to complete.

In the given exercise, the string’s wave velocity is 100 cm/s. This velocity affects how fast the wave's shape moves through the medium. It's important to note wave velocity depends on the string’s properties, like tension and mass per unit length.

Wavelength

Wavelength is the distance between two consecutive similar points on the wave, such as crest to crest or trough to trough. It can be calculated using the formula: \[ \lambda = \frac{v}{f} \]where \(v\) is the wave velocity and \(f\) is the frequency.

• For traveling waves in the exercise, the wavelength is determined to be 10 cm.
• Standing waves, however, form when two waves of the same frequency and amplitude travel in opposite directions, creating nodes where the string doesn't seem to move.

Understanding wavelength helps in analyzing the wave’s size and how its energy is propagated through the medium.

Standing Waves

Standing waves are fascinating because they appear to stand still. They occur when two waves of equal frequency and amplitude travel in opposite directions and interfere, creating distinct nodes (points of no motion) and antinodes (points of maximum motion).

• Nodes are points on the medium that remain at rest.
• Antinodes are points where the medium reaches its maximum amplitude.

In our exercise, the phase differences between points set up a potential standing wave. With points oscillating 180 degrees out of phase, the resulting interference pattern suggests the presence of nodes and antinodes, characteristics of standing waves.

Traveling Waves

Traveling waves move continuously along the medium, transporting energy from one point to another. Unlike standing waves, where certain points remain still, traveling waves have all points in motion but with different amplitudes and phases.

• They are described by a progressing wave pattern.
• For example, a wave on a string travels in one direction (right or left), spreading the wave energy through the medium.

In our scenario, the presence of multiple points and phase differences suggests a more complex pattern involving both traveling and standing wave components. This combination can be typical when waves interact constructively and destructively, leading to a dynamic yet predictable wave behavior.