Problem 64 - University Physics with Modern Physics

Question

A continuous succession of sinusoidal wave pulses are produced at one end of a very long string and travel along the length of the string. The wave has frequency 70.0 \(\mathrm{Hz}\) , amplitude 5.00 \(\mathrm{mm}\) , and wavelength 0.600 \(\mathrm{m} .\) (a) How long does it take the wave to travel a distance of 8.00 \(\mathrm{m}\) along the length of the string? (b) How long does it take a point on the string to travel a distance of \(8.00 \mathrm{m},\) once the wave train has reached the point and set it into motion? (c) In parts (a) and (b), how does the time change if the amplitude is doubled?

Step-by-Step Solution

Answer

(a) 0.190 s; (b) Calculate based on cycles; (c) Doubling amplitude affects point's time.

Step 1: Determine the Velocity of the Wave

The velocity \( v \) of a wave is calculated using the formula \( v = f \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength. Here, \( f = 70.0 \, \text{Hz} \) and \( \lambda = 0.600 \, \text{m} \). Substitute these values to get \( v = 70.0 \, \text{Hz} \times 0.600 \, \text{m} = 42.0 \, \text{m/s} \).

Step 2: Calculate Time Taken for Wave to Travel 8.00 m

Use the formula \( t = \frac{d}{v} \), where \( d \) is the distance and \( v \) is the wave velocity. Substitute \( d = 8.00 \, \text{m} \) and \( v = 42.0 \, \text{m/s} \) from the previous step to get \( t = \frac{8.00}{42.0} \approx 0.190 \text{ s} \).

Step 3: Calculate Time for a Point to Travel 8.00 m

A point on the string moves vertically in simple harmonic motion. The time taken for one complete wave cycle is the period \( T = \frac{1}{f} \). Here, \( T = \frac{1}{70.0} \approx 0.0143 \text{ s} \). The point covers a distance of one wavelength, \( 4 \times \text{amplitude} \), per cycle when considering its vertical motion. To cover 8.00 m vertically, calculate the number of cycles needed and multiply by \( T \).

Step 4: Compare Effects of Doubling Amplitude on Time

Doubling the wave's amplitude does not affect its velocity or the time it takes to traverse the string, as velocity depends only on frequency and wavelength. However, doubling the amplitude affects the point's motion, increasing the distance it travels per wave cycle, which can reduce the number of cycles needed to cover 8.00 m vertically.

Key Concepts

Sinusoidal Waves

Sinusoidal waves are a specific type of wave that exhibit a smooth, repetitive oscillation. They are called 'sinusoidal' because their shapes mimic the sine function, showing up as smooth, continuous curves.

These waves are essential because they model many waveforms in both physics and engineering. Here's how to recognize them and their characteristics:

• **Amplitude**: This is the height of the wave from the midpoint to the peak or trough. It affects the energy carried by the wave but not its speed.
• **Frequency**: This defines how many wave cycles pass a point in one second, measured in hertz (Hz).
• **Wavelength**: The distance between two consecutive crests or troughs. It dictates the wave's spatial repetition.

Sinusoidal waves are perfect for modeling harmonic oscillations such as sound waves and light waves. Understanding these elements helps in manipulating and interpreting waves in various scientific contexts.

Wave Velocity

Wave velocity is crucial for understanding how quickly a wave travels through a medium. Calculating this speed involves the frequency and wavelength of the wave: \[ v = f \lambda \]

Here, **\(v\)** represents the velocity, **\(f\)** is the frequency, and **\(\lambda\)** is the wavelength. Let's break down this powerful equation:

• **Higher frequency** means more cycles per second, thus generally faster movement.
• **Longer wavelength** can imply that the wave covers more distance per cycle.

Wave velocity doesn't change with amplitude. Whether measuring sound across air or light through space, this formula remains standard. Understanding velocity aids in applications like predicting how quickly signals travel in communication systems.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory motion of a point connected to a sinusoidal wave. This motion is periodic and is the foundation for understanding many physical systems such as springs and pendulums.

Characteristics of SHM include:

• **Cycles**: The point moves through a complete cycle continuously until external effects alter its path.
• **Period**: The time it takes to complete one full cycle, calculated as \(T = \frac{1}{f}\).

During SHM, the point follows the pattern of the sinusoidal wave vertically. While the distance traveled by the point during one cycle is influenced by its amplitude, the time taken for each cycle strictly depends on the frequency.

In the context of wave motion on a string, understanding SHM aids in calculating how long a point takes to move a specified distance, considering its vertical oscillations. Doubling the amplitude of the wave increases the vertical distance covered in one cycle, influencing the cumulative time required for specific motions.