To answer this question, we need to consider the concept of rotational motion. The rotational period of an object on a rotating system is the time it takes for that object to complete one full revolution around the axis of rotation.
For objects rotating in a circular path, all points on the same rigid body, such as a merry-go-round, have the same angular velocity \(\omega\), which is the rate of rotation measured in radians per second. However, the linear speed \(v\) of an object moving in a circular path does depend on its distance \(r\) from the axis of rotation, described by the equation:
\[ v = r \omega \]
Where \(v\) is the linear speed, \(r\) is the distance from the axis, and \(\omega\) is the angular velocity.
Given Jason is at a distance \(R\) and Betsy at a distance \(2R\), both will have the same angular velocity. This means they both complete their rotations in relation to the same measurement of time in terms of degrees or radians.
However, to find the rotational period \(T\) for each child, we can express it as:
\[ T = \frac{2\pi}{\omega} \]
Since \(\omega\) is a constant for both Jason and Betsy (they are on the same merry-go-round), the rotational period \(T\) would be the same for both regardless of their distances from the axis of rotation.
In conclusion, the rotational period of Jason is equal to the rotational period of Betsy.
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