The angular acceleration $\alpha$ of an object can be found using the following kinematic equation for rotational motion:

$\alpha = \frac{\omega_f - \omega_i}{t}$

where:

• $\omega_f = 3.68 \, \text{rad/s}$ is the final angular velocity,
• $\omega_i = 9.75 \, \text{rad/s}$ is the initial angular velocity,
• $t = 1.27 \, \text{s}$ is the time interval.

Now, substituting the given values into the equation:

$\alpha = \frac{3.68 \, \text{rad/s} - 9.75 \, \text{rad/s}}{1.27 \, \text{s}} = \frac{-6.07 \, \text{rad/s}}{1.27 \, \text{s}}$

$\alpha = -4.78 \, \text{rad/s}^2$

Thus, the angular acceleration is $-4.78 \, \text{rad/s}^2$, which indicates that the object is decelerating.

Would you like more details or have any questions about this?

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Here are some related questions to deepen your understanding:

1. What is the relationship between angular velocity, angular acceleration, and time?
2. How would the problem change if the object were speeding up instead of slowing down?
3. Can you derive this equation from the basic principles of rotational motion?
4. How do angular acceleration and tangential acceleration relate in circular motion?
5. What would the total angular displacement of the object be during this motion?

Tip: When angular acceleration is negative, it means the object is slowing down or decelerating.