As a new teacher, Emily is asked to play the national anthem on a violin before the first day of school. Just prior to taking the stage, she prepares by tuning a string on her violin to an A note (440 Hz).

Given:
- The vibrating section of the string is 0.50 m long.
- The whole string has a mass of 35 grams.
- The total length of the string is 0.69 m.

Determine the needed tension to tune the string so that she doesn't get "booed" by her new students.

Question

High School

Answer :

AsherIsaiah AsherIsaiah · Answer 1 · 2024-08-25T05:37:28-04:00

The needed tension to tune the string is approximately 96 Newtons.

To determine the needed tension to tune the string, we can use the formulas related to tension in a vibrating string.

First, let's calculate the mass per unit length of the string:

Mass per unit length = Total mass of the string / Total length of the string

Mass per unit length = 35 grams / 0.69 m

Mass per unit length = 50.72 grams/m

Next, let's calculate the velocity of waves on the string:

Velocity = sqrt(Tension / Mass per unit length)

Since we are given the vibrating section of the string is 0.50 m long, we can use this length to calculate the velocity:

Velocity = sqrt(Tension / 50.72 grams/m)

Now, we can use the given frequency of the A note (440 Hz) to calculate the velocity:

440 Hz = Velocity / (2 * 0.50 m)

Velocity = 440 Hz * 2 * 0.50 m

Velocity = 440 m/s

Finally, we can calculate the tension:

Tension = (Mass per unit length) * [tex](Velocity)^2[/tex]

Tension = 50.72 grams/m * (440 m/s)^2

Tension = 50.72 grams/m * [tex]193,600 m^2/s^2[/tex]

Tension = [tex]9,808,192 grams*m/s^2[/tex]

Converting grams to Newtons (1 gram = 0.0098 Newtons), the tension is approximately 96 Newtons.

Learn more about tension in a vibrating string here:

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