To determine which domains provide a real value for the period [tex]\( T \)[/tex] of a pendulum, we start with the given equation:
[tex]\[ T = 2\pi \sqrt{\frac{L}{g}} \][/tex]
Here:
- [tex]\( T \)[/tex] is the period of the pendulum.
- [tex]\( L \)[/tex] is the length of the string in meters.
- [tex]\( g \)[/tex] is the acceleration due to gravity in [tex]\( \text{m}/\text{s}^2 \)[/tex].
We need to analyze the expression inside the square root, [tex]\(\frac{L}{g}\)[/tex], to ensure that it yields a real number.
1. Domain [tex]\( g < 0 \)[/tex]:
For [tex]\( g < 0 \)[/tex], the expression [tex]\(\frac{L}{g}\)[/tex] becomes negative because [tex]\( L \)[/tex] is always positive (as it is a length). The square root of a negative number results in an imaginary number, which means the period [tex]\( T \)[/tex] would not be real.
Therefore, [tex]\( g < 0 \)[/tex] is invalid.
2. Domain [tex]\( g = 0 \)[/tex]:
For [tex]\( g = 0 \)[/tex], the expression [tex]\(\frac{L}{g}\)[/tex] becomes undefined because division by zero is not allowed in mathematics. Since this would result in an undefined operation, the period [tex]\( T \)[/tex] cannot be determined and is thus not real.
Therefore, [tex]\( g = 0 \)[/tex] is invalid.
3. Domain [tex]\( g > 0 \)[/tex]:
For [tex]\( g > 0 \)[/tex], the expression [tex]\(\frac{L}{g}\)[/tex] is positive because both [tex]\( L \)[/tex] and [tex]\( g \)[/tex] are positive. Taking the square root of a positive number results in a real number, which means the period [tex]\( T \)[/tex] will be real.
Therefore, [tex]\( g > 0 \)[/tex] is valid.
4. Domain [tex]\( g \geq 0 \)[/tex]:
For [tex]\( g \geq 0 \)[/tex], this includes both [tex]\( g > 0 \)[/tex] and [tex]\( g = 0 \)[/tex]. As analyzed above, [tex]\( g > 0 \)[/tex] is valid while [tex]\( g = 0 \)[/tex] is invalid because it leads to an undefined operation. Since this domain includes an invalid case, the entire domain [tex]\( g \geq 0 \)[/tex] cannot ensure a real value for the period.
Therefore, [tex]\( g \geq 0 \)[/tex] is invalid.
In summary, the valid domain that ensures a real value for the period [tex]\( T \)[/tex] of the pendulum is:
[tex]\[ g > 0 \][/tex]
