Calculating the New Frequency of a Stretched String

When the length of a stretched string is shortened by 20% and the tension is increased by 21%, the frequency of the string changes. To calculate the new frequency, follow the steps below:

Step 1: Calculate the new length of the string.

When the length of the string is shortened by 20%, it means that the new length is 80% of the original length. Therefore:

New length = 0.8 x original length

Step 2: Calculate the new tension in the string.

When the tension in the string is increased by 21%, it means that the new tension is 121% of the original tension. Therefore:

New tension = 1.21 x original tension

Step 3: Calculate the new frequency of the string.

The frequency of a stretched string depends on its length, tension, and mass per unit length. Since the mass per unit length of the string is assumed to be constant, the new frequency can be calculated using the following formula:

New frequency = (1/2L) * sqrt(T/u)

where L is the new length of the string, T is the new tension in the string, and u is the mass per unit length of the string.

Substituting the values obtained in steps 1 and 2 into the formula above, we get:

New frequency = (1/2 x 0.8 x original length) * sqrt(1.21 x original tension/u)

Simplifying the expression, we get:

New frequency = (0.4 * sqrt(1.21) * sqrt(original tension/u)) * sqrt(original length)

Step 4: Calculate the percentage change in frequency.

To calculate the percentage change in frequency, we need to compare the new frequency to the initial frequency (n). Therefore:

Percentage change in frequency = ((New frequency - n)/n) x 100%

Substituting the expression for new frequency obtained in step 3, we get:

Percentage change in frequency = (((0.4 * sqrt(1.21) * sqrt(original tension/u)) * sqrt(original length)) - n)/n) x 100%

Simplifying the expression, we get:

Percentage change in frequency = (0.4 * sqrt(1.21) * sqrt(original tension/u) * sqrt(original length)/n - 1) x 100%

Therefore, to calculate the new frequency of a stretched string that has been shortened by 20% and whose tension has been increased by 21%, you need to follow the four steps above.