91Ó°ÊÓ

Angular acceleration is a key concept in rotational motion. It measures how quickly the angular velocity of an object changes over time. Just as linear acceleration refers to the change in linear velocity, angular acceleration refers to the change in angular velocity. It is denoted by the symbol \( \alpha \).

• In this problem, the angular acceleration is given as \(+6.70\,\text{rad/s}^2\).
• This positive value indicates that the angular velocity is increasing in the counterclockwise direction.

Angular acceleration can affect how fast an object spins, making it crucial for understanding rotational dynamics. When calculating problems involving angular motion, knowing the angular acceleration is vital since it directly affects both angular velocity and angular displacement.

Kinematic equations are used to describe the motion of objects, whether they are moving in a straight line or rotating. For rotational motion, these equations allow us to calculate angular displacement, velocity, and acceleration.The key kinematic equation used in this exercise is:\[ \theta = \omega_i \cdot t + \frac{1}{2} \cdot \alpha \cdot t^2 \]

• \( \theta \) is the angular displacement.
• \( \omega_i \) is the initial angular velocity.
• \( t \) is the time of acceleration.
• \( \alpha \) is the angular acceleration.

Each of these terms plays a role in calculating how far an object rotates over a period of time. Once all the values are known, plugging them into the equation helps us find the angular displacement, as was done in the problem.

The initial angular velocity \(\omega_i\) is the starting rotational speed of an object before it experiences any acceleration. It is the counterpart in rotational motion to initial linear velocity.In this scenario, the initial angular velocity is necessary to solve the equation for angular displacement.

• The initial angular velocity was unknown at first but calculated using the equation \( \omega_f = \omega_i + \alpha \cdot t \)

By rearranging this equation, we can determine:\[ \omega_i = \omega_f - \alpha \cdot t \]Given:

• \( \omega_f = 74.5 \, \text{rad/s} \)
• \( \alpha = 6.70 \, \text{rad/s}^2 \)
• \( t = 4.50 \, \text{s} \)

Substituting in the values provides \( \omega_i = 44.35 \, \text{rad/s} \). Knowing this value is key to later finding the total angular displacement.

Final angular velocity \( \omega_f \) is the rotational speed of an object at the end of the acceleration period. It is calculated after the angular acceleration has occurred for a specified time.In the exercise:

• \( \omega_f = +74.5 \, \text{rad/s} \)

This indicates the speed each wheel reached after the 4.50 seconds of acceleration.
The final angular velocity is pivotal for calculating the initial angular velocity and for determining the effect of acceleration over time. This value shows how much the wheel's rotation increased, reflecting the change brought about by the angular acceleration. These calculations illustrate how understanding both the initial and final angular velocities are essential for analyzing motions in rotational dynamics.