# How Does Tension Affect the Frequency of Standing Waves?

• tomasblender
In summary, the arrangement shown in the figure involves a mass hung from a string connected to a vibrator of constant frequency f. The length of the string between point P and the pulley is 1.95 m, with a linear mass density of μ=0.00182 kg/m. Standing waves are observed when the mass is either 16.5 kg or 22.5 kg, but not with any mass in between. To find the frequency of the vibrator and the largest mass for which standing waves could be observed, equations for frequency and wavelength are used, resulting in different fundamental frequencies due to different tensions. To find the fundamental frequency, the wavelength is written in terms of m, u, and f, as well as L
tomasblender

## Homework Statement

In the arrangement shown in the figure, a mass can be hung from a string (with a linear mass density of μ=0.00182 kg/m) that passes over a light pulley. The string is connected to a vibrator (of constant frequency f), and the length of the string between point P and the pulley is L=1.95 m. When the mass m is either 16.5 kg or 22.5 kg, standing waves are observed; however no standing waves are observed with any mass between these values.

http://capa.physics.mcmaster.ca/figures/sb/Graph18/sb-pic1825.png

What is the frequency of the vibrator? (Hint: The greater the tension in the string the smaller the number of nodes in the standing wave.)

What is the largest mass for which standing waves could be observed?

f(m) = mf1
f = (T/u)^.5/2L

## The Attempt at a Solution

(16.5*9.8/0.00182)^.5/(2*1.95) = 79.249 Hz = (m + 1)f
(22.5*9.8/0.00182)^.5/(2*1.95) = 89.249 Hz = (m) f

Therefore

79.249 - 89.249 = mf + f - mf
10.000 Hz

would be

(mg/u)^.5/2L = Answer for part A
("a"^2)*(2L^2)*u/g = m

I cannot test if this is the answer without having the answer for the first section. Anyways I am dry out of ideas for part A, any help?

Last edited by a moderator:
tomasblender said:
f(m) = mf1
f = (T/u)^.5/2L

## The Attempt at a Solution

(16.5*9.8/0.00182)^.5/(2*1.95) = 79.249 Hz = (m + 1)f
(22.5*9.8/0.00182)^.5/(2*1.95) = 89.249 Hz = (m) f

You assumed that the fundamental frequency is the same in both scenarios. That's not true, since the tensions are different, resulting in different fundamental frequencies.

Try writing out the wavelength for both cases as a function of m, u, and f. Also write out the wavelength in terms of L and n1/n2 (representing the n1/n2th harmonic). Then you'll be able to find n1 and n2, which will tell you f.

## What is a standing wave?

A standing wave is a type of wave that occurs when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This creates a pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) that appear to be standing still.

## How is tension related to standing waves?

The tension of a medium, such as a string or rope, affects the speed at which a wave travels. As tension increases, the speed of the wave increases. This means that standing waves with higher tension will have a higher frequency and shorter wavelength.

## How do you calculate the wavelength of a standing wave?

The wavelength of a standing wave can be calculated by dividing the length of the medium by the number of nodes present in the wave. This is known as the fundamental frequency or first harmonic. The wavelength of higher harmonics can be calculated by dividing the length of the medium by the corresponding harmonic number.

## What is the relationship between standing waves and harmonics?

Harmonics are frequencies that are multiples of the fundamental frequency. In standing waves, the fundamental frequency is also known as the first harmonic. Higher harmonics can be created by adding nodes to the standing wave pattern, resulting in a greater number of antinodes. Each harmonic has its own wavelength and frequency.

## What are some practical applications of standing waves using tension?

Standing waves using tension are used in many musical instruments, such as guitars and violins, to produce different notes and harmonics. They are also important in radio and television broadcasting, as antennas use standing waves to transmit and receive signals. In addition, standing waves are used in medical imaging techniques, such as ultrasound, to create images of internal body structures.

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