A merry-go-round starts from rest and accelerates with angular acceleration \(0.010 \mathrm{rad} / \mathrm{s}^{2}\) for \(14 \mathrm{s}\). (a) How many revolutions does it make during this time? (b) What's its average angular speed?

Angular velocity is a measure of how quickly an object rotates or revolves around a central axis point. It is the rotational equivalent of linear velocity and can be understood as the angle an object sweeps out per unit time. For the merry-go-round mentioned in our exercise, this indicates how fast it spins around its center.

Mathematically, angular velocity, represented as \(\omega\), is often measured in radians per second (rad/s). To understand radians, consider a full circle, which accounts for \(2\pi\) radians – translating this to a merry-go-round, once it completes a full spin, it has swept through \(2\pi\) radians.

To calculate angular velocity, we can use the equation \(\omega = \omega_0 + \alpha t\), where \(\omega_0\) is the initial angular velocity and \(\alpha\) represents angular acceleration. As the merry-go-round starts from rest, its initial angular velocity \(\omega_0 = 0\), making the calculation straightforward once we know the angular acceleration and time.

Angular displacement refers to the change in the angle as an object rotates, in this case, how much the merry-go-round spins from its starting position. It's crucial to distinguish this from distance in linear motion, as angular displacement is measured in radians (or degrees), not meters.

The formula to find angular displacement \(\theta\) in rotational motion is akin to the formula for distance in linear motion, adapted to account for rotation: \(\theta = \omega_0 t + 0.5 \alpha t^2\). Following along with our example, since the merry-go-round begins at rest (\(\omega_0 = 0\)), the formula simplifies, and we only need to consider the acceleration and time squared.

Converting to Revolutions

To relate angular displacement to something more tangible like revolutions, you divide the radians by \(2\pi\). In the exercise, the merry-go-round's angular displacement was calculated to be 0.98 radians, which equates to approximately 0.156 revolutions.

Rotational motion is the movement of an object around a central axis. This axis can be internal, like the spinning of the Earth on its axis, or external, such as a merry-go-round rotating around its center support. The principles governing rotational motion closely parallel those for linear motion, with terms like velocity, displacement, and acceleration being translated into their angular counterparts.

In rotational motion, instead of tracking the path along a straight line, we account for the arc of the rotation, measuring in angles and radians. This is why understanding concepts like angular velocity and angular displacement is essential for problems involving rotation.

When addressing rotational motion problems, such as with the merry-go-round, the use of radians and consideration of the relationship between angular and linear quantities are crucial. The merry-go-round accelerates with a certain angular acceleration resulting in a final angular velocity, and over time, it sweeps out an angular displacement, completing a certain number of revolutions.