TG EAMCET 3 May 2025 Shift 1 Paper

Step 1: Formula for Fundamental Frequency
The basic formula for the frequency of a vibrating string is:
f=‌

1

2l

√‌

T

µ

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=‌

1

2l

√‌

T1

µ

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 ‌

3

4

l. The new tension is T2 and the new frequency becomes 2f1 (it increases by 100% ):
2f1=‌

1

2(‌ 3 4 l)

√‌

T2

µ

Step 4: Set Up the Equation
Now, make it easier by expressing 2f1 in terms of the formula:
2f1=‌

1

(3∕2)l

√‌

T2

µ

Because splitting the denominator gives 2×‌

1

2(‌ 3 4 l)

=‌

1

(3∕2)l

.
Step 5: Plug f1 from Step 2 into the New Equation
Replace f1 in the new formula using the old frequency formula:
2(‌

1

2l

√‌

T1

µ

)=‌

1

(3∕2)l

√‌

T2

µ

Step 6: Simplify the Equation
Multiply both sides to clear out the denominators:

‌

1

l

√‌

T1

µ

=‌

2

3l

√‌

T2

µ

Multiply both sides by 3l :
3√‌

T1

µ

=2√‌

T2

µ

Step 7: Get Rid of Square Roots by Squaring Both Sides
Square each side to eliminate the square roots:
9‌

T1

µ

=4‌

T2

µ

The µ cancels out:
9T1=4T2
Step 8: Find ‌

T2

T1

Divide both sides by 4T1 to solve for ‌

T2

T1

:
‌

T2

T1

=‌

9

4