TG EAMCET 3 May 2025 Shift 2 Paper

Step 1: Pendulum Frequency FormulaThe time it takes for one swing (period) of a simple pendulum is given by $T = 2\pi \sqrt{\frac{l}{g}}$, where $l$ is length and $g$ is gravity.Frequency ( $f$ ) is how many swings it makes in one second, so $f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{g}{l}}$.Step 2: Relationship Between Frequency and LengthFrom this, frequency is inversely related to the square root of length: $f \propto \frac{1}{\sqrt{l}}$.This means $\frac{f_2}{f_1} = \sqrt{\frac{l_1}{l_2}}$, where $l_1$ and $f_1$ are initial length and frequency, and $l_2$ and $f_2$ are final.Step 3: Assign the ValuesThe number of swings in a minute is frequency per minute. Original frequency $=72$, new frequency $=90$.So, $\frac{f_2}{f_1} = \frac{90}{72} = \frac{5}{4}$.Now, $\frac{5}{4} = \sqrt{\frac{l_1}{l_2}}$.Step 4: Find the New LengthSquare both sides: $\left(\frac{5}{4}\right)^2 = \frac{l_1}{l_2}$, so $\frac{25}{16} = \frac{l_1}{l_2}$.Rearrange to get $l_2 = \frac{16}{25} l_1$.Step 5: Calculate the Percentage DecreasedDecrease in length $= l_1 - l_2$.Percentage decrease $= \frac{l_1 - l_2}{l_1} \times 100 \%$.Substituting values: $\frac{l_1 - \frac{16}{25}l_1}{l_1} \times 100\% = \frac{9}{25} \times 100\% = 36\%$.