To analyze the relationship between the time period [tex]\( T \)[/tex] of a simple pendulum and the gravitational field [tex]\( g \)[/tex], let's start by looking at the given formula:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{g}} \][/tex]
We will break this down step by step:
1. Understanding the Formula: The formula [tex]\( T = 2 \pi \sqrt{\frac{L}{g}} \)[/tex] indicates that the time period [tex]\( T \)[/tex] depends on both the length of the pendulum [tex]\( L \)[/tex] and the gravitational field [tex]\( g \)[/tex].
2. Rewriting the Formula: To analyze how [tex]\( T \)[/tex] is related to [tex]\( g \)[/tex], we can rewrite the formula to better see the dependency:
[tex]\[ T = 2 \pi \sqrt{L} \cdot \sqrt{\frac{1}{g}} \][/tex]
3. Isolating the Dependency on [tex]\( g \)[/tex]: From the rewritten formula, it’s clear that [tex]\( T \)[/tex] involves [tex]\( g \)[/tex] in the form of [tex]\( \sqrt{\frac{1}{g}} \)[/tex]. Thus, we can represent this as:
[tex]\[ T ∝ \sqrt{\frac{1}{g}} \][/tex]
4. Interpreting the Proportional Relationship: This means that [tex]\( T \)[/tex] is directly proportional to the inverse of the square root of [tex]\( g \)[/tex]. In mathematical terms:
[tex]\[ T \propto \frac{1}{\sqrt{g}} \][/tex]
5. Conclusion: Therefore, the statement [tex]\( T \propto g \)[/tex] is incorrect. From our analysis, the correct proportionality relationship is:
[tex]\[ T \propto \frac{1}{\sqrt{g}} \][/tex]
In summary, the time period [tex]\( T \)[/tex] of a simple pendulum is proportional to the inverse of the square root of the gravitational field [tex]\( g \)[/tex], and not directly proportional to [tex]\( g \)[/tex].
