# Exploring Standing Waves on a Vibrating String: Determining Linear Density

• limeater
In summary, the problem involves a string of fixed length L=1.200m being vibrated at a fixed frequency of f=120.0Hz, with a variable tension Ts. Standing waves with fewer than seven nodes are observed on the string when the tension is 2.654N and 4.147N, but not for any intermediate tension. To find the linear density of the string, the equation for the natural frequency of the string is used, along with the relation v=sqrt(T/u), where u is the linear density. Two methods are suggested, one involving finding two values of u that match for different modes and tensions, and the other involving using the ratio of modes to find the linear density.
limeater
A string of fixed length L=1.200m is vibrated at a fixed frequency of f=120.0Hz. The tension, Ts, of the string can be varied. Standing waves with fewer than seven nodes are observed on the string when the tension is 2.654N and 4.147N, but not for any intermediate tension. What is the linear density of the string?

Hey. this I've been trying to understand this question for over half an hour now with no luck. what does it mean with "fewer than 7 nodes".
Any help is greatly appreciated

yeah i understand what nodes are and also looked through the link. I also understand that a standing wave can exist on a string only if its wavelength can be given by the equation:

wavelength= 2 x length of string / m
m being an integer >0 and also the mode of the string

the mode in this case can be anything between 1 and 8
im still lost though

Well from the link it gives you the equation for the natural frequency of a string including the mass which is unknown. everything else is apart from the mode of oscillation. Its easy to modify the equation to dea with other modes. Then you just play around and see which two modes with the two tensions give you the same value for mass.

Also remember linear density is m/L.

ok so would this be a correct way of approaching it?

by combining f=mv/2L and v=srqt(T/u)
m being the mode, T being the tension and u being the linear density
we can derive u=(T)(m^2)/(2Lf)^2
with the values given we have
u=(T)(m^2)/82944

Now I make a table

------T=2.654N-------T=4.147N
m=1
m=2
.
.
.
m=8

Try and match two values of u on the two columns and I will have my answer?

Yeah you can do it that way.

i thought of another way of doing it which relates to the chart

consider x and y to be 2 different modes between 1 and 8

then
(2.654)(x^2)=(4.147)(y^2)
x/y=srt(4.147/2.654)
x/y=1.25
x/y=5/4
meaning that the linear density is equal at mode 5 and tension 2.654 to the linear density at mode 4 and tension 4.147
linear density=8.0x10^-4

The problem is that x/y is not exactly equal to 1.25
its more like 1.25002, so will my method be still correct?

Thats how I did it.

## 1. What are standing waves?

Standing waves are a type of wave that occurs when two waves with the same frequency and amplitude travel in opposite directions, intersect and interfere with each other. This results in a wave pattern that appears to be standing still, hence the name "standing wave".

## 2. How are standing waves formed?

Standing waves are formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This interference causes certain points along the wave to have no displacement, creating nodes, while other points have maximum displacement, creating antinodes.

## 3. Where can standing waves be found?

Standing waves can be found in a variety of natural and man-made systems. Some examples include musical instruments, such as guitar strings and organ pipes, as well as electromagnetic waves in transmission lines and even ocean waves between a wave and a reflecting surface.

## 4. What is the importance of standing waves?

Standing waves have many practical applications in various fields of science and technology. They are used in musical instruments to produce specific tones and in antennas to improve signal strength. They are also helpful in studying the properties of waves and understanding the behavior of different wave systems.

## 5. How are standing waves different from traveling waves?

The main difference between standing waves and traveling waves is that standing waves do not propagate through space, while traveling waves do. Standing waves appear to be stationary, while traveling waves move through a medium. Additionally, standing waves have fixed nodes and antinodes, while traveling waves do not have fixed points of maximum and minimum displacement.

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