91Ó°ÊÓ

Angular acceleration is a measure of how quickly an object's rotational speed changes. It tells us how fast the angular velocity increases or decreases over time. In the given problem, the merry-go-round starts from rest, meaning its initial angular velocity, \( \omega_0 \), is zero.
The formula for angular acceleration, \( \alpha \), is \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the time period over which the change occurs.

• Given final angular velocity: \( \omega = 0.68 \, \text{rad/s} \)
• Time interval: \( t = 24 \, \text{s} \)

Since the initial velocity \( \omega_0 = 0 \), the change in angular velocity is equal to the final velocity. Thus, the angular acceleration is \( \alpha = \frac{0.68}{24} \approx 0.02833 \, \text{rad/s}^2 \).
Understanding angular acceleration helps us figure out how much the merry-go-round speeds up as time progresses.

The moment of inertia describes how the mass of an object is distributed relative to its axis of rotation. It is crucial in understanding how much torque is needed to change the rotational speed of the object.
For a disk-like object such as the merry-go-round, the moment of inertia, \( I \), can be calculated using the formula:
\[ I = \frac{1}{2} M r^2 \]

• Mass \( M = 31,000 \, \text{kg} \)
• Radius \( r = 7.0 \, \text{m} \)

Substitute these values into the formula to find the moment of inertia:
\[ I = \frac{1}{2} \times 31,000 \times 7.0^2 = 759,500 \, \text{kg} \cdot \text{m}^2 \]
The moment of inertia tells us how resistant the merry-go-round is to changes in its rotation and plays a vital role in calculating the net torque.

A uniform disk, like the merry-go-round in this scenario, has its mass evenly distributed across its circular shape. This uniformity simplifies calculations as the formula for the moment of inertia specifically applies due to its symmetric mass distribution.
When analyzing problems involving rotations, assuming a uniform disk simplifies figuring out parameters like moment of inertia and, consequently, torque requirements.

• The symmetry means the rotational dynamics only depend on total mass and radius.
• Such assumptions save time and reduce complexity in mathematical computations.

Given these properties, understanding that the merry-go-round behaves as a uniform disk helps streamline the process of calculating torque, allowing one to employ specific formulas confidently.
It's important to verify when an object can be approximated as a uniform disk to apply these calculations correctly.