Tension

To determine the ratio of the new tension to the original tension for the vibrating string, we should consider the fundamental frequency formula for a stretched string:

Where:

• is the frequency of vibration.
• is the length of the string.
• is the tension in the string.
• is the linear mass density of the string, assumed constant in both cases.

Initially, the frequency is 200 Hz, and the length is .

When the string length is doubled to and the tension changes, the frequency becomes Hz.

Using the formula for frequency in both scenarios, we have:

Divide the second equation by the first equation:

After simplification:

By solving the frequency ratio, we find:

Now, solve for :

Squaring both sides yields:

Hence, the ratio of the new tension to the original tension is 9:1.