A 250-kg weight is hanging from the ceiling by a thin copper...

A $250-\mathrm{kg}$ weight is hanging from the ceiling by a thin copper wire. In its fundamental mode, this wire vibrates at the frequency of concert $\mathrm{A}(440 \mathrm{Hz}) .$ You then increase the temperature of the wire by $40 \mathrm{C}^{\circ} .$ (a) By how much will the fundamental frequency change? Will it increase or decrease? (b) By what percentage will the speed of a wave on the wire change? (c) By what percentage will the wavelength of the fundamental standing wave change? Will it increase or decrease?

Key Concepts

Vibrating String Fundamentals

The fundamental mode of vibration in a string is characterized by the relation f = v / (2L), where f is the frequency, v is the wave speed, and L is the length of the string. This concept is central to understanding how factors such as tension and mass per unit length affect the natural frequencies of a string.

Wave Speed on a String

The speed at which waves propagate along a string is given by v = ?(T/?), where T is the tension in the string and ? is its linear mass density. Changes in tension—due to mechanical or thermal effects—therefore directly influence the wave speed and subsequently the vibrational frequencies.

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in length when subjected to a change in temperature. In the context of a vibrating wire, an increase in temperature typically results in an increase in length and a corresponding decrease in tension (if the force applied remains constant), which in turn affects the wave speed and the fundamental frequency.

Small Percentage Change Analysis

When examining the effects of small temperature changes on wave properties, it is useful to perform a relative or percentage change analysis. This involves understanding how small variations in tension or length, as dictated by material properties such as the coefficient of thermal expansion, lead to corresponding small percentage changes in wave speed, frequency, and wavelength.

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