A 12 kg hanging sculpture is suspended by a 90-cm -long, 5.0 g steel wire. When the wind blows h

step 1 Hi everyone, this is the problem based on formation of standing waves over the stretch strength.

A 12 kg hanging sculpture is suspended by a 90-cm -long, 5.0...

A 12 kg hanging sculpture is suspended by a $90-\mathrm{cm}$ -long, $5.0 \mathrm{g}$ steel wire. When the wind blows hard, the wire hums at its fundamental frequency. What is the frequency of the hum?

Key Concepts

Tension in a Suspended Mass

This concept pertains to the force exerted by a hanging mass due to gravity. When an object is suspended by a string, the tension in the string is equal to the weight of the object (mg). This tension acts as the restoring force for any vibrations in the string, and is a critical factor in determining the behavior of waves along it.

Linear Mass Density

Linear mass density is defined as the mass per unit length of a string or wire. It is a key parameter in wave mechanics because it influences how quickly disturbances travel along the string. In vibration problems, knowing the linear mass density allows for the calculation of the wave speed, which is essential for determining the frequencies of the vibrations.

Wave Speed on a String

The speed of a wave traveling along a stretched string is determined by the tension in the string and its linear mass density. The relationship is given by the equation v = sqrt(T/?), where v is the wave speed, T is the tension, and ? is the linear mass density. This speed is pivotal for understanding the behavior of standing waves on the string.

Fundamental Frequency of a Vibrating String

The fundamental frequency is the lowest natural frequency at which a string vibrates and is determined by the wave speed and the length of the string. It can be calculated using the formula f = 1/(2L) * sqrt(T/?). This formula shows that the frequency is directly related to the square root of the tension and inversely proportional to both the linear mass density and the length of the string.

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