To determine the correct relationship between the linear speed \( v \) and the angular speed \( \omega \) of a body moving uniformly in a circle of radius \( \gamma \), let's examine each option carefully:
When a body is moving uniformly in a circular path, its linear speed \( v \) and angular speed \( \omega \) are related by the radius of the circle, \( \gamma \).
1. Option A: \( v = \omega \gamma \)
- This equation states that the linear speed is the product of the angular speed and the radius of the circle. This is a standard formula used in circular motion, where:
[tex]\[
v = \omega \gamma
\][/tex]
2. Option B: \( v = \omega^2 \gamma \)
- This equation implies that the linear speed is the product of the square of the angular speed and the radius. However, this does not hold true in the context of circular motion.
3. Option C: \( v = \omega \gamma^2 \)
- This equation suggests that the linear speed is the product of the angular speed and the square of the radius. This is not correct for the relationship between linear and angular speed in circular motion.
4. Option D: \( v^2 = \omega \gamma \)
- This equation implies that the square of the linear speed is the product of the angular speed and the radius. This is not the standard formula used in circular motion dynamics.
5. Option E: \( v = \frac{\omega}{\gamma} \)
- This equation suggests that the linear speed is the angular speed divided by the radius. This does not correctly describe the relation between these quantities in uniform circular motion.
The correct relationship is given by Option A:
[tex]\[
v = \omega \gamma
\][/tex]
Thus, the correct option is [tex]\( \boxed{A} \)[/tex].
