Math Snap

STEP 1

1. The radius of the circle is $4 \, \text{m}$.
2. The ball completes the circle in $1.5 \, \text{minutes}$.
3. We need to find the angular speed in radians per second.

STEP 2

1. Calculate the circumference of the circle.
2. Determine the time in seconds.
3. Calculate the angular speed.

STEP 3

The circumference of a circle is given by the formula $C = 2 \pi r$, where $r$ is the radius. Here, the radius $r = 4 \, \text{m}$.
$$C = 2 \pi \times 4$$ $$C = 8 \pi \, \text{m}$$

STEP 4

Convert the time from minutes to seconds. We know that $1 \, \text{minute} = 60 \, \text{seconds}$.
$$1.5 \, \text{minutes} = 1.5 \times 60 \, \text{seconds}$$ $$1.5 \, \text{minutes} = 90 \, \text{seconds}$$

Angular speed is defined as the angle in radians covered per unit time. Since the ball covers the entire circumference, it travels $2\pi$ radians in $90 \, \text{seconds}$.
$$\text{Angular speed} = \frac{2\pi \, \text{radians}}{90 \, \text{seconds}}$$ $$\text{Angular speed} = \frac{\pi}{45} \, \text{radians per second}$$ The angular speed of the ball is:
$$\boxed{\frac{\pi}{45} \, \text{radians per second}}$$