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A physics student suspends a 1 kg mass from a pulley using a wire as shown in Figure \(12.39 .\) The pulley allows her to easily change the length of the wire. In order to determine the speed of a transverse wave on the wire, she measures the frequency of the fundamental mode as a function of the length of the wire \(L\). The table shows her results: $$\begin{array}{lc}\hline L(\mathrm{~m}) & f(\mathrm{H} z) \\\\\hline 1.0 & 48 \\\0.5 & 98 \\\0.3 & 160 \\\0.2 & 249 \\\\\hline\end{array}$$ Make two plots of these data. First plot the frequency as a function of \(L_{\text {, and then plot it as a function of } 1 / L \text { . Which graph shows a }}\) linear relationship? Draw a best-fit line through the linear graph's data, and use the slope of this line to determine the wave speed.

Step by step solution

01

Understand the Relationship

The frequency of the fundamental mode of a wave on a string is related to the length of the string. The frequency \( f \) is inversely related to the length \( L \) of the string, meaning \( f \propto \frac{1}{L} \). This implies that plotting \( f \) against \( \frac{1}{L} \) should yield a linear graph.

02

Prepare the Data

Calculate the reciprocal of lengths \( \frac{1}{L} \) for each given value of \( L \): - For \( L = 1.0 \) m, \( \frac{1}{L} = 1.0 \) m鈦宦�- For \( L = 0.5 \) m, \( \frac{1}{L} = 2.0 \) m鈦宦�- For \( L = 0.3 \) m, \( \frac{1}{L} 鈮� 3.33 \) m鈦宦�- For \( L = 0.2 \) m, \( \frac{1}{L} = 5.0 \) m鈦宦�

03

Plot Frequency vs Length

Plot the given frequency \( f \) values against the wire length \( L \): - (1.0, 48)- (0.5, 98)- (0.3, 160)- (0.2, 249)This plot typically does not show a linear relationship given the data set.

04

Plot Frequency vs Reciprocal Length

Plot the frequency \( f \) against \( \frac{1}{L} \) values:- (1.0, 48)- (2.0, 98)- (3.33, 160)- (5.0, 249)This plot should produce approximately a straight line, indicating a linear relationship.

05

Determine the Slope of the Linear Plot

Fit a best-fit line through the points on the frequency versus \( \frac{1}{L} \) plot.Calculate the slope \( m \) of this line. Use points \((x_1, y_1) = (1.0, 48)\) and \((x_2, y_2) = (5.0, 249)\) to estimate:\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{249 - 48}{5 - 1} = 50.25\] Hz m.

06

Calculate the Wave Speed

The wave speed \( v \) is determined using the relationship \( v = 2L \times f \). Rearranging terms for a string vibrating in its fundamental mode gives \( v = 2 \times \text{slope}\). The speed of the wave on the wire is:\( v = 2 \times 50.25 = 100.5 \) m/s.

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Key Concepts

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

Fundamental Frequency

The fundamental frequency, often referred to as the first harmonic, is the lowest frequency at which a system naturally vibrates. It is a crucial concept when analyzing waves on strings, especially in physics experiments like the one described in the exercise. When a string is plucked or struck, it begins vibrating at this fundamental frequency, producing a pitch or tone. The fundamental frequency depends on several factors: the length of the string, its tension, and its mass per unit length. In simpler terms, altering any of these factors will change the fundamental frequency. For example, increasing the tension will raise the frequency, while lengthening the string will lower it.

String Length and Wave Relationship

Understanding the interplay between string length and wave behavior is essential in wave mechanics. The length of a string is inversely related to the frequency of the wave it produces. This means that as the length of the string increases, the frequency of the wave decreases, and vice versa. A practical view of this can be observed in musical instruments. A guitarist shortens the string by pressing down on a fret to increase the pitch, effectively raising the wave frequency. The mathematical expression of this relationship in the exercise is shown as \( f \propto \frac{1}{L} \), where \( f \) is the frequency, and \( L \) is the string length.

Inverse Proportionality

Inverse proportionality is a mathematical relationship where one quantity increases while another decreases. In the given exercise, the frequency and the length of the string are inversely proportional. This relationship can be expressed with \( f \propto \frac{1}{L} \), which implies that as the length of the string declines, the frequency escalates. This concept is crucial in physics and engineering because it allows us to predict how changes in one variable will affect another. If students create a graph plotting \( f \) against \( \frac{1}{L} \), the data should reveal a straight line, highlighting the linear nature facilitated by inverse proportionality.

Best-Fit Line

Drawing a best-fit line is a fundamental technique in data analysis, used to identify the underlying trend of a data set. In the context of the exercise, plotting the frequency against the reciprocal of length and fitting a best-fit line helps to visualize the linear relationship driven by inverse proportionality. The slope of this line is particularly important as it is used to estimate the wave speed. By selecting appropriate data points, students can calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This slope is then doubled to find the wave speed, linking the graphical analysis to a physical characteristic of the wave, enhancing comprehension of both mathematical and physical applications.