To solve the problem, we will use equations of rotational motion:

1. Initial Phase (t = 0 to t = 3.00 s):

   • Initial angular velocity ($\omega_0$) = 27.0 rad/s
   • Angular acceleration ($\alpha_1$) = 31.0 rad/s²
   • Time ($t_1$) = 3.00 s

   Angular velocity at t = 3.00 s:

   $\omega_1 = \omega_0 + \alpha_1 t$ $\omega_1 = 27.0 + (31.0)(3.00) = 27.0 + 93.0 = 120.0 \, \text{rad/s}$

   Angular displacement during this phase:

   $\theta_1 = \omega_0 t + \frac{1}{2} \alpha_1 t^2$ $\theta_1 = (27.0)(3.00) + \frac{1}{2}(31.0)(3.00)^2$ $\theta_1 = 81.0 + \frac{1}{2}(31.0)(9.00) = 81.0 + 139.5 = 220.5 \, \text{rad}$

2. Second Phase (Deceleration to a stop):

   • Final angular displacement = 441 rad
   • Initial angular velocity for this phase = $\omega_1 = 120.0 \, \text{rad/s}$
   • Final angular velocity = 0 (stopped)
   • Angular displacement ($\theta_2$) = 441 rad
   • Use the equation: $\omega^2 = \omega_0^2 + 2\alpha \theta$ Solve for $\alpha_2$: $0 = (120.0)^2 + 2\alpha_2 (441)$ $14400 = -2\alpha_2 (441)$ $\alpha_2 = -\frac{14400}{2(441)} = -\frac{14400}{882} = -16.33 \, \text{rad/s}^2$

   Time taken to decelerate ($t_2$):

   Using the equation: $\omega = \omega_0 + \alpha_2 t$ $0 = 120.0 + (-16.33)t_2$ $t_2 = \frac{120.0}{16.33} = 7.35 \, \text{s}$

3. Total Results: a) Total angle rotated ($\theta_{\text{total}}$): $\theta_{\text{total}} = \theta_1 + \theta_2$ $\theta_{\text{total}} = 220.5 + 441 = 661.5 \, \text{rad}$

   b) Total time ($t_{\text{total}}$): $t_{\text{total}} = t_1 + t_2$ $t_{\text{total}} = 3.00 + 7.35 = 10.35 \, \text{s}$

   c) Deceleration ($\alpha_2$): $\alpha_2 = -16.33 \, \text{rad/s}^2$

Final Answers:

a) Total angle rotated: $661.5 \, \text{rad}$
b) Total time: $10.35 \, \text{s}$
c) Deceleration: $-16.33 \, \text{rad/s}^2$

Let me know if you'd like further clarifications or detailed explanations.

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5 Related Questions:

1. What is the average angular velocity of the wheel during its entire motion?
2. How much energy is dissipated during the deceleration phase, assuming the wheel has a moment of inertia $I$?
3. What is the total distance traveled by a point at the rim of the wheel (if the radius is known)?
4. How would the results change if the acceleration were not constant?
5. How can these equations be adapted for non-rotational (linear) motion?

Tip: Always verify your kinematic equations by double-checking unit consistency and physical interpretations of the results.