A mix work of art/musical tool is illustrated. 6 pieces of similar piano wire (cut to different lengths) space hung native the same support, andmasses are hung indigenous the complimentary end of every wire. Each cable is 1, 2, or 3 systems long, and also each supports 1, 2, or 4 units of mass. The fixed of each cable is negligiblecompared to the total mass hanging indigenous it. As soon as a strong breeze blows, the wires vibrate and create one eerie sound.

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Question:

Rank each wire-mass device on the basis of its basic frequency.Rank from biggest to smallest. To rank items as equivalent, overlap them.


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General guidance


Concepts and reason

The concepts used to settle this problem are standing wave on string.

First use the relation in between the cable tension, cable mass, and string length to calculate the fundamental frequencies for every the wire-mass system.

Finally compare them on your basis of fundamental frequencies and rank them biggest to smallest.


Fundamentals

Expression for the anxiety in the spring is,

*

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

Expression because that the wave velocity is,

*

Here,

*
is the mass density which consistent for every the cases of the wire-mass system,
*
is the tide velocity, and also
*
is the length of the string.

Substitute

*
for the T.

*

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

Expression for the basic frequency is,

*

Here,

*
is the basic frequency.

Substitute

*
because that v.

*


Step-by-step


Step 1 of 2


The fundamental frequency

*
of wire-mass mechanism A is given as follows:

*

Substitute 2L for

*
and 2m because that
*
in the above equation.

*

Substitute f because that

*
in the above equation.

*

Fundamental frequency because that the wire-mass system B which has actually 2 devices of mass and also 1 unit of size is,

The fundamental frequency

*
of wire-mass system B is offered as follows:

*

Substitute L for

*
and 2m because that
*
in the above equation.

*

Substitute f because that

*
in the over equation.

*

The basic frequency

*
of wire-mass device C is provided as follows:

*

Substitute 2L for

*
and m for
*
in the over equation.

*

Substitute f for

*
in the over equation.

*


Explanation | usual mistakes | hints for next step

From the expression

*
, clearly shows the the fundamental frequency is directly proportional to
*
.

Mass density is consistent for every the cases and the stress and anxiety becomes same to the weights of the offered masses. For this reason the fundamental frequency for the wire-mass device is dependence on the mass and also length the the wire.


The not correct expression to calculation the fundamental frequency

*
because the correct expression is
*
.


Compare each an essential frequency and rank lock on basis of its an essential frequency from biggest to smallest.


Step 2 that 2


The fundamental frequency

*
of wire-mass system D is provided as follows:

*

Substitute 3L for

*
and m for
*
in the above equation.

*

Substitute f for

*
in the above equation.

*

Fundamental frequency because that the wire-mass device E which has 4 units of mass and 2 systems of length is,

The an essential frequency

*
of wire-mass system E is given as follows:

*

Substitute 2L for

*
and 4m because that
*
in the over equation.

*

Substitute f for

*
in the above equation.

*

The basic frequency

*
of wire-mass system F is given as follows:

*

Substitute L because that

*
and m for
*
in the over equation.

*

Substitute f for

*
in the over equation.

*

The ranking from largest to smallest is,

*


Wire-mass device ranked from largest to smallest on your basis of its an essential frequency as follows

*


Wave speed is proportional its stress and anxiety which is nothing yet weight of the mass. An essential wavelength is proportional come the size of the cable whereas its fixed of the cable is negligible.

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

As the length increases an essential frequency also increases.


The incorrect expression to calculate the an essential frequency

*
since the correct expression is
*
.


Answer


Wire-mass device ranked from largest to smallest on their basis the its an essential frequency as follows

*


Answer only


Wire-mass mechanism ranked from biggest to the smallest on your basis the its fundamental frequency as follows

*


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