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Angular velocity is a core concept in circular motion that describes how fast an object rotates or revolves around a central point. In this scenario, the merry-go-round completes one full revolution in 4 seconds. It's essential to understand that a full revolution equals a movement of 2\(\pi\) radians, which is the angular distance for a circular path.
To calculate angular velocity \( \omega \), we divide the total angular displacement by the time taken for one complete cycle. Hence, \( \omega = \frac{2\pi}{4.0} = \frac{\pi}{2} \text{ radians/second}\). This measure gives us the rate of rotation and is crucial for finding other aspects of motion, like linear speed and centripetal acceleration. Remember that angular velocity is a vector quantity, with direction typically along the axis of rotation according to the right-hand rule.

Linear speed in circular motion is the path distance traveled per unit of time. It can be envisioned as how fast a child moves along the edge of the merry-go-round.
The formula to link linear speed \(v\) with angular velocity is \(v = r \cdot \omega\), where \(r\) is the radius (distance from the rotation axis to the edge).

• The radius in this scenario is 1.2 meters.
• The angular velocity, as calculated previously, is \(\frac{\pi}{2} \text{ rad/s}\).

Thus, \(v = 1.2 \times \frac{\pi}{2} = 1.2\pi \text{ m/s} \). This equation illustrates how the linear speed increases with either a longer radius or faster angular velocity. In simpler terms, the further out you are from the center, the faster you have to travel to complete one revolution in the same time.

Centripetal acceleration is a key concept in the study of circular motion that refers to the acceleration directed towards the center of the circle along which the body is moving. This acceleration keeps the object in circular motion rather than moving in a straight line.
It can be calculated using the formula \( a_c = \omega^2 \cdot r \). This equation indicates that centripetal acceleration depends on both the square of the angular velocity and the radius of the circle.
In our problem:

• The angular velocity \(\omega = \frac{\pi}{2} \text{ rad/s}\).
• The radius \(r = 1.2 \text{ m}\).

Substituting these values, we find \( a_c = \left(\frac{\pi}{2}\right)^2 \times 1.2 = \frac{\pi^2}{4} \times 1.2 \approx 2.96 \text{ m/s}^2\).
This resultant acceleration keeps the child securely moving in a circle, always pointing inward toward the center of rotation. This type of acceleration is constant in magnitude but continuously changes direction as the child rides around the merry-go-round.