Problem 50 - General Chemistry: Principles and Modern Applications

Question

What is the length of a string that has a standing wave with four nodes (including those at the ends) and \(\lambda=17 \mathrm{cm} ?\)

Step-by-Step Solution

Answer

The length of the string is 51 cm.

Step 1: Determine the number of wavelengths

Since a standing wave with two nodes equals one wavelength, then a standing wave with four nodes will contain three wavelengths (four nodes minus one). Therefore, there are 3 wavelengths in the string.

Step 2: Calculate the length of the string

Once the number of wavelengths is known, the length of the string can be calculated by multiplying this number by the length of one wavelength. This is because the length of the string constitutes three wavelengths in our case. Therefore, the length of the string can be calculated as \(3 * \lambda\). With \( \lambda = 17 \, cm\), the length of the string will be \(3 * 17 \, cm = 51 \, cm\).

Key Concepts

Nodes

Nodes are points in a standing wave where there is no movement. These are the quiet spots in a wave pattern, indicating zero amplitude. For a string fixed at both ends, nodes are always formed at the ends.
In the problem provided, a string has four nodes, including the ones at each end. This means that inside the string, the vibrational pattern must have two additional nodes.

• Start by understanding the relationship: nodes occur at intervals of half a wavelength.
• The presence of four nodes suggests there are three segments on the string, each half-wavelength long.

This idea is key to understanding the structure of standing waves. Each segment between nodes is effectively half a wavelength. Thus, more nodes mean more segments, which is crucial for calculating wavelength.

Wavelength

Wavelength is the distance over which the wave's shape repeats. In this case, one complete wave pattern is described as \( \lambda=17 \mathrm{cm} \). Understanding this helps in determining how many wavelengths fit into a given string.
The length of string segments can be linked to the wavelength:

• Each segment between nodes is half the wavelength.
• When the string supports a mode with four nodes, it means the full string can accommodate multiple half-wavelength segments.

Wavelength plays a vital role in measurements concerning nodes and wave behavior. By knowing the wavelength and node spacing, we can break down the standing wave pattern into understandable segments.

Wave Calculation

Wave calculation involves understanding how wavelength and nodes determine the standing wave pattern length. From the solution given, the number of wavelengths is calculated:

• Given four nodes, subtract one to find the number of full waves. So, 3 wavelengths fit the string.

The formula to find the length: \[ \text{Length of string} = \text{Number of waves} \times \lambda \]
By substituting the values:

• 3 full waves \( \times 17 \, \mathrm{cm} \ = 51 \, \mathrm{cm} \)

This demonstrates how fully understanding nodes and wavelength relation simplifies practical wave calculations. Remember, recognizing node patterns and wavelength values is essential for finding the correct standing wave formation.