The centripetal acceleration $a_c$ of an object moving in a circular path is given by the formula:

$a_c = \frac{v^2}{r}$

Where:

• $v$ is the tangential speed of the object, and
• $r$ is the radius of the circular path.

We can calculate the tangential speed $v$ using the formula:

$v = \frac{2\pi r}{T}$

Where:

• $r$ is the radius of the circular path,
• $T$ is the time for one complete rotation (period), and
• $2\pi r$ is the circumference of the circular path.

Step 1: Find the radius

The diameter of the carousel is 4.0 meters, so the radius is half of that:

$r = \frac{4.0}{2} = 2.0 \, \text{m}$

Step 2: Calculate the tangential speed

The period $T$ is 6 seconds, so we can calculate the tangential speed $v$:

$v = \frac{2\pi \times 2.0}{6} = \frac{4\pi}{6} \approx 2.09 \, \text{m/s}$

Step 3: Calculate the centripetal acceleration

Now that we have $v$, we can find the centripetal acceleration:

$a_c = \frac{v^2}{r} = \frac{(2.09)^2}{2.0} \approx \frac{4.37}{2.0} \approx 2.19 \, \text{m/s}^2$

Final Answer:

The centripetal acceleration is approximately $2.19 \, \text{m/s}^2$.

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Would you like further details or have any questions? Here are some related questions to think about:

1. How would the centripetal acceleration change if the carousel rotated faster?
2. What would the effect be if the diameter of the carousel were doubled?
3. How is centripetal force related to centripetal acceleration?
4. What would happen if the girl moved closer to the center of the carousel?
5. How does centripetal acceleration compare to gravitational acceleration?

Tip: Remember that the period and radius are key in calculating both the speed and the centripetal acceleration of any rotating object!