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The gravitational constant, commonly denoted as \( g \), plays a crucial role in pendulum motion. It represents the acceleration due to gravity acting on objects near Earth's surface. Typically, \( g \) is approximately \( 9.81 \, \text{m/s}^2 \). In the context of a pendulum, this constant helps determine how fast the pendulum swings back and forth.
Understanding this constant is important because it influences the period \( T \) of a pendulum. A higher gravitational constant would mean a shorter period, as the pendulum would swing faster under greater gravitational pull. Conversely, a lower \( g \) would extend the period.
The gravitational constant helps to standardize calculations of pendulum behavior, allowing us to make predictions about the period based on varying lengths \( l \). Without it, comparing pendulum behavior across different environmental conditions would be difficult, since \( g \) can vary slightly with elevation and geographical location.

Percentage change is a useful concept for understanding how alterations in one variable affect another proportionally. In the pendulum scenario, we're interested in how a change in period \( T \) affects the pendulum's length \( l \).
To express a percentage change, you compare the difference between the new and the original value, then divide by the original value, and multiply by 100 to convert to a percentage. For example, a 30% increase in \( T \) implies \( T' = 1.3T \).
Calculating the corresponding change in length \( l \) involves following the same process. Using the relationship established earlier, if the new length is 1.69 times the initial length, the percentage increase in length is \( 69\% \). This tells us that for a 30% increase in the period, the length of the pendulum increases by 69%.

The relationship between length \( l \) and period \( T \) of a pendulum is crucial for understanding pendulum dynamics. According to the formula \( T = 2 \pi \sqrt{\frac{l}{g}} \), the period \( T \) is directly related to the square root of the length \( l \).
This means that as \( l \) increases, \( T \) also increases, but not linearly. Instead, it follows a square root relationship; doubling the length of the pendulum, for instance, would result in a period that is roughly \( \sqrt{2} \) or about 1.41 times longer.
When tackling problems like the one presented, where an increase in the period by 30% results in an increased length, it's this non-linear square root function that explains the larger-than-expected increase in length. This relationship demonstrates why small changes in period can lead to significant changes in length.