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.

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.