How does the fundamental frequency ($f$) change if the length ($L$) of the string is doubled while tension remains constant?

The frequency is halved. According to the relationship $f \propto \frac{1}{L}$, doubling the length increases the wavelength of the first harmonic, which inversely reduces the frequency for a constant wave speed.

What is the difference between the mass used to calculate tension and the mass used to calculate $\mu$?

The mass for tension ($T=mg$) is the hanging weight stretching the string. The mass for $\mu$ is the actual mass of the string material itself divided by its length.

Why is a signal generator preferred over a simple mechanical oscillator in this practical?

A signal generator allows for very fine adjustments in frequency (high resolution). This is necessary to precisely locate the 'sharp' peak of resonance for the first harmonic.

What error occurs if the string is not perfectly horizontal between the vibration generator and the pulley?

The measured length $L$ may not accurately represent the vibrating segment, and the tension component might not be purely vertical. This leads to inaccuracies in the calculated wave speed.

Define 'Mass per unit length' ($\mu$) and state its SI units.

It is the ratio of the string's total mass to its total length, representing the linear density of the medium. Its SI units are $kg\,m^{-1}$.

When plotting $f$ against $\sqrt{T}$, what does the gradient represent?

The gradient represents $\frac{1}{2L\sqrt{\mu}}$. This is derived from rearranging $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ into the form $y = mx$.

How do you identify the first harmonic visually during the experiment?

The string will vibrate with a single large 'loop' or envelope. There will be no points of zero movement (nodes) except at the two fixed ends.

Why should the signal generator be left to stabilize for several minutes before taking readings?

Electronic components can drift in frequency as they warm up. Stabilization ensures the frequency output remains constant during the measurement of the resonant peak.

Compare the phase of two points located on the same 'loop' of a stationary wave.

All points within a single loop (between two adjacent nodes) are in phase. They all reach their maximum positive and negative displacements at the same time.

What is the relationship between wave speed ($v$) and tension ($T$)?

Wave speed is proportional to the square root of tension ($v \propto \sqrt{T}$). If tension is quadrupled, the wave speed doubles.

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5. Waves & Particle Nature of Light

Core Practical 7: Investigating Stationary Waves

Summary

This practical investigation explores the relationship between the fundamental frequency of a stationary wave on a stretched string and its physical properties: length, tension, and mass per unit length. By manipulating these variables, the wave speed can be determined and the theoretical wave equation $v = \sqrt{\frac{T}{\mu}}$ can be verified.

1. Definition & Core Concepts

Stationary Waves: These are formed by the superposition of two progressive waves of the same frequency and amplitude traveling in opposite directions. In this practical, the waves are generated by a vibration generator and reflected at a fixed boundary.
Fundamental Frequency ( $f_0$ ): Also known as the first harmonic, this is the lowest frequency at which a stationary wave forms, characterized by a single antinode in the center and nodes at the fixed ends.
Nodes and Antinodes: Nodes are points of zero displacement where destructive interference occurs, while antinodes are points of maximum displacement where constructive interference occurs.

Diagram of a stationary wave experimental setup showing a vibration generator, a string under tension from hanging masses, and the first harmonic wave pattern.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions