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The problem presented in your image is asking for the angle generated by point \( P \) on a circle given its radius, angular velocity, and time.

### Given:
- \( r = 10 \) cm (radius of the circle)
- \( \omega = \frac{\pi}{6} \) radians per second (angular speed)
- \( t = 4 \) seconds (time)

### (a) What is the angle \( \theta \) generated by point \( P \) in time \( t \)?

To calculate the angle \( \theta \) in radians, we use the formula:

\[
\theta = \omega \cdot t
\]

### Substituting the values:
\[
\theta = \left(\frac{\pi}{6} \, \text{radians/sec}\right) \times 4 \, \text{sec} = \frac{4\pi}{6} = \frac{2\pi}{3} \, \text{radians}
\]

Thus, the angle \( \theta \) generated by point \( P \) is \( \frac{2\pi}{3} \) radians.

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Would you like more details or further clarification on this?  
Here are 5 related questions for practice:

1. How does the radius affect the angle generated by a point in circular motion?
2. If the angular velocity was doubled, what would the angle be after 4 seconds?
3. What is the linear distance traveled by point \( P \) along the circle in this time?
4. How would the result change if the time was 6 seconds instead of 4?
5. Can you convert \( \frac{2\pi}{3} \) radians into degrees?

### Tip:
Always ensure that the units of angular velocity and time are compatible when calculating angular displacement.

Suppose that point P is on a circle with radius r, and ray OP is rotating with angular speed ω. Complete parts (a). r = 10 cm, ω = π/6 radian per sec, t = 4 sec. (a) What is the angle generated by P in time t?

This problem explores the concept of angular displacement in circular motion. With a circle of radius 10 cm, angular velocity of π/6 rad/sec, and time duration of 4 seconds, you will calculate the angle generated by point P. Learn how to apply the formula θ = ω * t and understand circular motion at an advanced level.

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