Answer :
To solve this problem, we need to identify the correct values for the magnitude of angular velocity ([tex]\(\omega\)[/tex]), linear velocity ([tex]\(\gamma\)[/tex]), and acceleration ([tex]\(a\)[/tex]) for a body rotating with uniform speed in a circle of radius [tex]\(T\)[/tex].
### Definitions and Formulas
1. Angular Velocity ([tex]\(\omega\)[/tex]):
The angular velocity is the rate at which an object rotates around a circle. If the period (time for one complete revolution) is [tex]\(T\)[/tex], then
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
2. Linear Velocity ([tex]\(\gamma\)[/tex]):
The linear velocity is the tangential speed of the object moving along the circle. It is related to angular velocity and radius [tex]\(r\)[/tex] by the formula:
[tex]\[ \gamma = r \cdot \omega = r \cdot \frac{2\pi}{T} \][/tex]
3. Centripetal Acceleration ([tex]\(a\)[/tex]):
The centripetal acceleration is given by:
[tex]\[ a = r \cdot \omega^2 = r \cdot \left(\frac{2\pi}{T}\right)^2 = r \cdot \frac{4\pi^2}{T^2} \][/tex]
Given that radius [tex]\(r = T\)[/tex] is assumed (as per the question's context), we can substitute [tex]\(r\)[/tex] with [tex]\(T\)[/tex] in the above formulas.
### Calculations
- Angular Velocity ([tex]\(\omega\)[/tex]):
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
- Linear Velocity ([tex]\(\gamma\)[/tex]):
[tex]\[ \gamma = T \cdot \frac{2\pi}{T} = 2\pi \][/tex]
- Centripetal Acceleration ([tex]\(a\)[/tex]):
[tex]\[ a = T \cdot \left(\frac{2\pi}{T}\right)^2 = T \cdot \frac{4\pi^2}{T^2} = \frac{4\pi^2}{T} \][/tex]
### Evaluating the Choices
Now, let's evaluate the options given based on our computations:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \omega & \gamma & a \\ \hline A) & \frac{\pi}{T} & \frac{4\pi T}{T} & \frac{2\pi T}{T^2} \\ \hline B) & \frac{2\pi}{2T} & \frac{2\pi T}{2T} & \frac{\pi^2}{T^2} \\ \hline C) & \frac{2\pi}{T} & \frac{2\pi}{T} & \frac{4\pi^2}{T^2} \\ \hline D) & \frac{2\pi}{T} & \frac{4\pi}{T} & \frac{4\pi^2}{T^2} \\ \hline \end{array} \][/tex]
By comparing the calculations:
- Option C correctly matches with our calculations for [tex]\(\omega\)[/tex], [tex]\(\gamma\)[/tex], and [tex]\(a\)[/tex]:
- [tex]\(\omega = \frac{2\pi}{T}\)[/tex]
- [tex]\(\gamma = 2\pi\)[/tex]
- [tex]\(a = \frac{4\pi^2}{T^2}\)[/tex]
Hence, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
### Definitions and Formulas
1. Angular Velocity ([tex]\(\omega\)[/tex]):
The angular velocity is the rate at which an object rotates around a circle. If the period (time for one complete revolution) is [tex]\(T\)[/tex], then
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
2. Linear Velocity ([tex]\(\gamma\)[/tex]):
The linear velocity is the tangential speed of the object moving along the circle. It is related to angular velocity and radius [tex]\(r\)[/tex] by the formula:
[tex]\[ \gamma = r \cdot \omega = r \cdot \frac{2\pi}{T} \][/tex]
3. Centripetal Acceleration ([tex]\(a\)[/tex]):
The centripetal acceleration is given by:
[tex]\[ a = r \cdot \omega^2 = r \cdot \left(\frac{2\pi}{T}\right)^2 = r \cdot \frac{4\pi^2}{T^2} \][/tex]
Given that radius [tex]\(r = T\)[/tex] is assumed (as per the question's context), we can substitute [tex]\(r\)[/tex] with [tex]\(T\)[/tex] in the above formulas.
### Calculations
- Angular Velocity ([tex]\(\omega\)[/tex]):
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
- Linear Velocity ([tex]\(\gamma\)[/tex]):
[tex]\[ \gamma = T \cdot \frac{2\pi}{T} = 2\pi \][/tex]
- Centripetal Acceleration ([tex]\(a\)[/tex]):
[tex]\[ a = T \cdot \left(\frac{2\pi}{T}\right)^2 = T \cdot \frac{4\pi^2}{T^2} = \frac{4\pi^2}{T} \][/tex]
### Evaluating the Choices
Now, let's evaluate the options given based on our computations:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \omega & \gamma & a \\ \hline A) & \frac{\pi}{T} & \frac{4\pi T}{T} & \frac{2\pi T}{T^2} \\ \hline B) & \frac{2\pi}{2T} & \frac{2\pi T}{2T} & \frac{\pi^2}{T^2} \\ \hline C) & \frac{2\pi}{T} & \frac{2\pi}{T} & \frac{4\pi^2}{T^2} \\ \hline D) & \frac{2\pi}{T} & \frac{4\pi}{T} & \frac{4\pi^2}{T^2} \\ \hline \end{array} \][/tex]
By comparing the calculations:
- Option C correctly matches with our calculations for [tex]\(\omega\)[/tex], [tex]\(\gamma\)[/tex], and [tex]\(a\)[/tex]:
- [tex]\(\omega = \frac{2\pi}{T}\)[/tex]
- [tex]\(\gamma = 2\pi\)[/tex]
- [tex]\(a = \frac{4\pi^2}{T^2}\)[/tex]
Hence, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
