Question

A combination work of art/musical instrument is illustrated. Six pieces of identical piano wire (cut to different lengths) are hung from the same support, andmasses are hung from the free end of each wire. Each wire is 1, 2, or 3 units long, and each supports 1, 2, or 4 units of mass. The mass of each wire is negligiblecompared to the total mass hanging from it. When a strong breeze blows, the wires vibrate and create an eerie sound.

Question:

Rank each wire-mass system on the basis of its fundamental frequency.Rank from largest to smallest. To rank items as equivalent, overlap them.

Answer 1

The concept used to solve this problem is standing wave on a string.

First use the relation between the string tension, string mass, and string length to calculate the fundamental frequencies for all the wire-mass systems.

Finally, compare them on their basis of fundamental frequencies and rank them largest to smallest.

Fundamentals

Expression for the tension in the spring is,

T=Mg

Here, T is the tension in the string, M is the hanging mass, and g is the acceleration due to gravity.

Expression for the wave velocity is,

Here, m/L is the mass density which is constant for all the cases of the wire-mass system, v is the wave velocity, and L is the length of the string.

Substitute Mg for the T.

The fundamental vibration mode of a stretched string is such that the wavelength is twice of the length of the string.

Expression for the fundamental frequency is,

Here, is the fundamental frequency.

Substitute for v.

The fundamental frequency f_A of wire-mass system A is given as follows:

Substitute 2L for L_A and 2m for m_A in the above equation.

Substitute f for in the above equation.

Fundamental frequency for the wire-mass system B which has 2 units of mass and 1 unit of length is,

The fundamental frequency f_B of wire-mass system B is given as follows:

Substitute L for L_B and 2m for m_B in the above equation.

Substitute f for in the above equation.

The fundamental frequency f_C of wire-mass system C is given as follows:

Substitute 2L for L_C and m for m_C in the above equation.

Substitute f for in the above equation.

Explanation | Common mistakes | Hint for next step

From the expression , clearly shows that the fundamental frequency is directly proportional to .

Mass density is constant for all the cases and the tension becomes equal to the weights of the given masses. Hence the fundamental frequency for the wire-mass system is dependent on the mass and length of the wire.

The incorrect expression to calculate the fundamental frequency because the correct expression is .

Compare each fundamental frequency and rank them on basis of its fundamental frequency from largest to smallest.

The fundamental frequency f_D of wire-mass system D is given as follows:

Substitute 3L for L_D and m for m_D in the above equation.

Substitute f for in the above equation.

Fundamental frequency for the wire-mass system E which has 4 units of mass and 2 units of length is,

The fundamental frequency f_E of wire-mass system E is given as follows:

Substitute 2L for L_E and 4m for m_E in the above equation.

Substitute f for in the above equation.

The fundamental frequency f_F of wire-mass system F is given as follows:

Substitute L for L_F and m for m_F in the above equation.

Substitute f for in the above equation.

The ranking from largest to smallest is,

Wire-mass system ranked from largest to smallest on their basis of its fundamental frequency as follows

Wave speed is proportional its tension which is nothing but the weight of the mass. Fundamental wavelength is proportional to the length of the string whereas its mass of the string is negligible.

Fundamental frequency is proportional to the square root of masses hung and inversely proportional to the length of the string.

As the length increases fundamental frequency also increases.

The incorrect expression to calculate the fundamental frequency because the correct expression is .

                 

Wire-mass system ranked from largest to smallest on their basis of its fundamental frequency as follows