Let's go over the problem based on the image.

You are asked to solve parts (a) through (c) involving a point $P$ rotating on a circle with given values for radius $r$, angular speed $\omega$, and time $t$.

Given:

• Radius $r = 10 \, \text{cm}$
• Angular speed $\omega = \frac{\pi}{6} \, \text{radians per second}$
• Time $t = 4 \, \text{seconds}$

(a) What is the angular displacement $\theta$ in time $t$?

The angular displacement $\theta$ is calculated using the formula: $\theta = \omega \cdot t$ Substituting the values: $\theta = \frac{\pi}{6} \times 4 = \frac{4\pi}{6} = \frac{2\pi}{3} \, \text{radians}$

(b) What is the distance traveled by point $P$ along the circle in time $t$?

The distance traveled $s$ along the circumference of a circle is given by: $s = r \cdot \theta$ Substituting $r = 10 \, \text{cm}$ and $\theta = \frac{2\pi}{3} \, \text{radians}$: $s = 10 \cdot \frac{2\pi}{3} = \frac{20\pi}{3} \, \text{cm}$

(c) What is the linear speed of point $P$?

The linear speed $v$ is related to the angular speed by: $v = r \cdot \omega$ Substituting $r = 10 \, \text{cm}$ and $\omega = \frac{\pi}{6} \, \text{radians/sec}$: $v = 10 \cdot \frac{\pi}{6} = \frac{10\pi}{6} = \frac{5\pi}{3} \, \text{cm/sec}$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the angular displacement change if the angular speed was doubled?
2. What is the relationship between angular displacement and distance traveled on a circular path?
3. If the radius was increased, how would the linear speed change for the same angular speed?
4. What is the formula for calculating the angular velocity in terms of the frequency of rotation?
5. How would the distance traveled change if the time $t$ was halved?

Tip: When working with circular motion, always distinguish between angular velocity (measured in radians per second) and linear velocity (measured in distance per second).