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

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Answer :

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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