Problem 75 - University Physics with Modern Physics

Question

In your physics lab, an oscillator is attached to one end of a horizontal string. The other end of the string passes over a frictionless pulley. You suspend a mass \(M\) from the free end of the string, producing tension \(Mg\) in the string. The oscillator produces transverse waves of frequency \(f\) on the string. You don't vary this frequency during the experiment, but you try strings with three different linear mass densities \(\mu\). You also keep a fixed distance between the end of the string where the oscillator is attached and the point where the string is in contact with the pulley's rim. To produce standing waves on the string, you vary \(M\); then you measure the node-to-node distance \(d\) for each standingwave pattern and obtain the following data: (a) Explain why you obtain only certain values of \(d\). (b) Graph \(\mu d^2\) (in kg \(\cdot\) m) versus \(M\) (in kg). Explain why the data plotted this way should fall close to a straight line. (c) Use the slope of the best straight-line fit to the data to determine the frequency \(f\) of the waves produced on the string by the oscillator. Take \(g = 9.80 \, \mathrm{m/s}^2\). (d) For string A (\(\mu = 0.0260\) g/cm), what value of \(M\) (in grams) would be required to produce a standing wave with a node-to-node distance of 24.0 cm? Use the value of \(f\) that you calculated in part (c).

Step-by-Step Solution

Answer

Only specific values of \(d\) produce standing waves; the graph of \(\mu d^2\) versus \(M\) is linear due to its proportional relationship; use the slope to find frequency \(f\). For string A, use \(f\) to determine \(M\) for the desired node-to-node distance.

Step 1: Understanding Standing Waves

Standing waves occur when waves reflect off a boundary and interfere constructively with incoming waves. This happens only at certain lengths where the conditions for constructive interference are met, leading to specific node-to-node distances, given by \( d = \frac{n\lambda}{2} \), where \( n \) is the harmonic number and \( \lambda \) is the wavelength.

Step 2: Relate Wavelength and Mass Density

The velocity \( v \) of a wave on a string is given by \( v = \sqrt{\frac{T}{\mu}} \) and the wave equation is \( v = f\lambda \). For standing waves, \( \lambda = \frac{2d}{n} \), so \( v = \frac{2df}{n} \). Combining these gives \( f = \frac{n}{2d}\sqrt{\frac{T}{\mu}} \). With tension \( T = Mg \), this relationship becomes \( \mu d^2 = \frac{n^2 g}{4f^2} M \), suggesting \( \mu d^2 \propto M \).

Step 3: Graphing \( \mu d^2 \) versus \( M \)

Plot \( \mu d^2 \) on the y-axis and \( M \) on the x-axis. According to the derived relationship, this graph should be a straight line with a slope \( m = \frac{n^2 g}{4f^2} \). With given data, you can find \( f \) by fitting the best line and calculating the slope.

Step 4: Calculating the Wave Frequency \( f \)

After obtaining the slope \( m \) from the graph, use the equation \( m = \frac{n^2 g}{4f^2} \) to find \( f \). Rearranging gives \( f = \frac{n\sqrt{g}}{2\sqrt{m}} \). Insert values for \( m \), \( n \), and \( g \) to solve for the wave frequency \( f \).

Step 5: Determine Mass \( M \) for Specific Conditions

With \( f \) known, determine the mass \( M \) needed to create a standing wave with \( d = 24.0 \) cm for string A. From \( \mu d^2 = \frac{n^2 g M}{4f^2} \), rearrange to find \( M = \frac{4f^2 \mu d^2}{n^2 g} \) and substitute \( \mu = 0.0260 \) g/cm and \( d = 24.0 \) cm.

Key Concepts

Wave Frequency

Wave frequency refers to how often the wave oscillates, or cycles, within a certain time frame in a stationary position. It is commonly measured in hertz (Hz), which equals one cycle per second. Understanding wave frequency is crucial in this exercise because it governs how the waves behave on the string. The oscillator in our setup vibrates at a constant frequency, setting the rhythm for these waves.

In the context of standing waves, each harmonic represents a specific frequency that leads to unique patterns of nodes and antinodes. These frequencies are determined by the length of the string and its properties such as tension and mass density. The equation that connects wave frequency with other properties of the wave on a string is fundamental to understanding how standing waves are formed and maintained.

Mass Density

Mass density, when discussing waves on strings, refers to the mass per unit length of the string, typically denoted by the symbol \( \mu \), with units like kg/m or g/cm. It plays a pivotal role because it directly affects the wave velocity. In our physics lab exercise, different strings with varying linear mass densities \( \mu \) are tested to develop a comprehensive understanding of how this variable impacts the formation of standing waves.

The tension in the string, a result of the hanging mass, combines with mass density to affect wave speed using the relationship \( v = \sqrt{\frac{T}{\mu}} \). Changes in mass density alter the velocity of the wave, influencing the conditions needed for standing wave formation. Therefore, having a firm grasp of mass density helps predict how waves will behave under different experimental setups and how they interact with other factors like tension and wave frequency.

Wave Equation

The wave equation ties together crucial aspects of wave behavior—velocity \( v \), frequency \( f \), and wavelength \( \lambda \). A fundamental relationship described by this equation is \( v = f\lambda \), providing a snapshot of how waves travel. The velocity of a wave on a string depends on its tension and mass density, described by \( v = \sqrt{\frac{T}{\mu}} \).

For the specific case of standing waves, the wavelengths and frequencies must match the string's natural modes. This leads to our simplified expression where \( \lambda = \frac{2d}{n} \) for the wave on a string, showing direct dependency on the node-to-node distance \( d \), which is determined by harmonic number \( n \). The interplay between these elements guides us to determine wave phenomena like standing wave patterns. As such, understanding and applying the wave equation is vital to solving the experimental challenge.

Constructive Interference

Constructive interference is a phenomenon where two or more waves superimpose to form a larger amplitude wave. This occurs when the crests and troughs of the interacting waves align perfectly, adding up rather than canceling out. In the setup given, constructive interference is essential for forming standing waves, which consist of nodes (points of zero amplitude) and antinodes (points of maximum amplitude).

In order to achieve standing waves, the wave frequency and wavelength must be such that specific lengths or node-to-node distances are established. This leads to the reflection and perfect overlap of the waves traveling back and forth along the string. The tension and mass density are critical factors to meet the conditions necessary for this type of interference. Understanding this concept provides insight into why only specific wave patterns can form on the string when experimenting in the laboratory.