Step 1: Formula for Fundamental Frequency
The basic formula for the frequency of a vibrating string is:
f=‌√‌Here,
f is frequency,
l is the length of the string,
T is the tension, and
µ is mass per unit length (which stays the same in this problem).
Step 2: Frequency with Initial Length and Tension
The first frequency, with length
l and tension
T1 is:
f1=‌√‌Step 3: Frequency After Changing Length and Tension
The length of the string is shortened by
25%, so the new length is
75% of
l, or
‌l. The new tension is
T2 and the new frequency becomes
2f1 (it increases by
100% ):
2f1=‌√‌Step 4: Set Up the Equation
Now, make it easier by expressing
2f1 in terms of the formula:
2f1=‌√‌Because splitting the denominator gives
2×‌=‌.
Step 5: Plug
f1 from Step 2 into the New Equation
Replace
f1 in the new formula using the old frequency formula:
2(‌√‌)=‌√‌Step 6: Simplify the Equation
Multiply both sides to clear out the denominators:
‌√‌=‌√‌Multiply both sides by
3l :
3√‌=2√‌Step 7: Get Rid of Square Roots by Squaring Both Sides
Square each side to eliminate the square roots:
9‌=4‌The
µ cancels out:
9T1=4T2Step 8: Find
‌Divide both sides by
4T1 to solve for
‌ :
‌=‌
