Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the domains that provide a real value for the period [tex]\(T\)[/tex] of a pendulum, let us analyze the given equation carefully:
[tex]\[ T = 2\pi \sqrt{\frac{L}{g}} \][/tex]
where:
- [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]\( \text{m/s}^2 \)[/tex]
### Analysis by Case:
1. Case [tex]\( g < 0 \)[/tex]
- When [tex]\( g \)[/tex] is negative, the term [tex]\( \frac{L}{g} \)[/tex] becomes negative.
- The square root of a negative number is not a real number (it becomes an imaginary number).
- Therefore, for [tex]\( g < 0 \)[/tex], the period [tex]\( T \)[/tex] is not a real value.
2. Case [tex]\( g = 0 \)[/tex]
- When [tex]\( g = 0 \)[/tex], the term [tex]\( \frac{L}{g} \)[/tex] involves division by zero, which is undefined in mathematics.
- Hence, for [tex]\( g = 0 \)[/tex], the period [tex]\( T \)[/tex] is undefined.
3. Case [tex]\( g > 0 \)[/tex]
- When [tex]\( g \)[/tex] is positive, the term [tex]\( \frac{L}{g} \)[/tex] is positive.
- The square root of a positive number is real.
- Therefore, for [tex]\( g > 0 \)[/tex], the period [tex]\( T \)[/tex] is a real value.
4. Case [tex]\( g \geq 0 \)[/tex]
- This domain includes both [tex]\( g > 0 \)[/tex] and [tex]\( g = 0 \)[/tex].
- As analyzed, for [tex]\( g = 0 \)[/tex] the period [tex]\( T \)[/tex] is undefined.
- For [tex]\( g > 0 \)[/tex], the period [tex]\( T \)[/tex] is real.
- Thus, the domain [tex]\( g \geq 0 \)[/tex] does not uniformly provide real values (it includes a case where the value is undefined).
### Conclusion:
The only domain that guarantees the period [tex]\( T \)[/tex] of the pendulum to be a real value is:
[tex]\[ g > 0 \][/tex]
Thus, the correct domain which provides a real value for the period is:
[tex]\[ g > 0 \][/tex]
[tex]\[ T = 2\pi \sqrt{\frac{L}{g}} \][/tex]
where:
- [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]\( \text{m/s}^2 \)[/tex]
### Analysis by Case:
1. Case [tex]\( g < 0 \)[/tex]
- When [tex]\( g \)[/tex] is negative, the term [tex]\( \frac{L}{g} \)[/tex] becomes negative.
- The square root of a negative number is not a real number (it becomes an imaginary number).
- Therefore, for [tex]\( g < 0 \)[/tex], the period [tex]\( T \)[/tex] is not a real value.
2. Case [tex]\( g = 0 \)[/tex]
- When [tex]\( g = 0 \)[/tex], the term [tex]\( \frac{L}{g} \)[/tex] involves division by zero, which is undefined in mathematics.
- Hence, for [tex]\( g = 0 \)[/tex], the period [tex]\( T \)[/tex] is undefined.
3. Case [tex]\( g > 0 \)[/tex]
- When [tex]\( g \)[/tex] is positive, the term [tex]\( \frac{L}{g} \)[/tex] is positive.
- The square root of a positive number is real.
- Therefore, for [tex]\( g > 0 \)[/tex], the period [tex]\( T \)[/tex] is a real value.
4. Case [tex]\( g \geq 0 \)[/tex]
- This domain includes both [tex]\( g > 0 \)[/tex] and [tex]\( g = 0 \)[/tex].
- As analyzed, for [tex]\( g = 0 \)[/tex] the period [tex]\( T \)[/tex] is undefined.
- For [tex]\( g > 0 \)[/tex], the period [tex]\( T \)[/tex] is real.
- Thus, the domain [tex]\( g \geq 0 \)[/tex] does not uniformly provide real values (it includes a case where the value is undefined).
### Conclusion:
The only domain that guarantees the period [tex]\( T \)[/tex] of the pendulum to be a real value is:
[tex]\[ g > 0 \][/tex]
Thus, the correct domain which provides a real value for the period is:
[tex]\[ g > 0 \][/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.