91Ó°ÊÓ

Rotational acceleration, denoted as \(\alpha\), represents how quickly the angular velocity of an object is changing over time. It is similar to linear acceleration, but applies to rotational motion. In our exercise, the disk starts from rest, meaning the initial angular velocity is zero. Using the kinematic equation that relates angular displacement \(\theta\), initial angular velocity \(\omega_0\), time \(t\), and rotational acceleration \(\alpha\), we can determine the acceleration. The simplified form becomes:
\[ \theta = \frac{1}{2}\alpha t^2 \]
By rearranging and solving for \(\alpha\), we find:
\[ 25 = \frac{1}{2}\alpha (5)^2 \]
\[\alpha = 2 \text{ rad/s}^2\]
This illustrates that the disk’s angular acceleration is 2 rad/s². An important point to remember is that rotational acceleration remains constant, allowing us to use these kinematic relations effectively.

Angular velocity, represented by \(\omega\), is a measure of how fast an object is rotating. It's akin to linear velocity but in a rotational framework. From our problem, we first determine the average angular velocity over the initial 5 seconds using:

\[ \overline{\omega} = \frac{\theta}{t} \]
Substituting the given values:

\[ \overline{\omega} = \frac{25}{5} = 5 \text{ rad/s} \]

Additionally, the instantaneous angular velocity at the end of the 5 seconds is found using:

\[ \omega = \omega_0 + \alpha t \]
Given that \(\omega_0 = 0\) and \(\alpha = 2 \text{ rad/s}^2\), we calculate:

\[ \omega = 0 + 2 \times 5 = 10 \text{ rad/s} \]

Hence, at the end of 5 seconds, the disk's angular velocity is 10 rad/s. Both average and instantaneous angular velocities provide insights into the disk’s rotational dynamics.

Kinematic equations are crucial tools for solving rotational motion problems. They describe how quantities like angular displacement \(\theta\), angular velocity \(\omega\), and rotational acceleration \(\alpha\) relate over time. In rotational motion, the primary kinematic equation we use is:

\[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \]

This equation helped us find the disk's rotational acceleration when starting from rest, confirmed by:

\[ 25 = \frac{1}{2}\alpha (5)^2 \]
\[\alpha = 2 \text{ rad/s}^2\]

Moreover, we used:
\[ \omega = \omega_0 + \alpha t \]

to determine the instantaneous angular velocity at a specific time. Finally, to find the angle turned in the next 5 seconds, we revisited the kinematic equation with updated initial conditions:

\[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \]

Substituting the new values, where \(\omega_0 = 10 \text{ rad/s}\) (as obtained previously), and \(\alpha = 2 \text{ rad/s}^2\), we computed:
\[ \theta = 10 \times 5 + \frac{1}{2} \times 2 \times 5^2 = 75 \text{ rad} \]

Grasping these kinematic relationships empowers you to tackle a variety of rotational motion challenges efficiently.