It looks like you have an AP Physics C problem set on rotational motion, involving angular impulse, angular momentum, angular speed, and power. Here’s an outline of each part in case you'd like any clarifications or details on solving each:

1. Angular Impulse (Part a):

   • This involves integrating the applied force function over time to get an expression for the angular impulse $J_{\text{rot}}$.
   • The applied force is $F(t) = \beta t^2 + \gamma$, and by integrating this force with respect to time from $t = 0$ to $t$, you should find $J_{\text{rot}}$ as a function of $t$, $\beta$, and $\gamma$.

2. Angular Momentum at $t = 2$ seconds (Part b):

   • Using the angular impulse found in part (a), you can determine the angular momentum $L$ at $t = 2$ seconds.
   • Since the object starts from rest, the angular momentum should equal the angular impulse up to that time.

3. Angular Speed After 3 Seconds (Part c.i):

   • Angular speed $\omega$ can be found by using $L = I \omega$, where $L$ is the angular momentum derived in previous steps and $I$ is the rotational inertia.

4. Translational Speed (Part c.ii):

   • For this, you'll use $v = R \omega$, where $R$ is the distance from the axis to the point where the force is applied.

5. Instantaneous Power at $t = 3$ seconds (Part c.iii):

   • Instantaneous power can be calculated with $P = \tau \omega$, where $\tau$ is the torque at $t = 3$ seconds and $\omega$ is the angular velocity at that time.

Would you like more detailed solutions or explanations for any specific part of this problem set?

Here are five related questions for further exploration:

1. How would the angular impulse change if the force was applied at a constant rate instead of a time-dependent one?
2. How does the rotational inertia $I$ affect the angular speed after a given time?
3. What additional steps would be required if the object had an initial angular velocity?
4. How would frictional torque impact the instantaneous power calculation at $t = 3$ seconds?
5. How could you calculate the total work done by the applied force over a specified time interval?

Tip: Remember to check units consistently, especially when dealing with time-dependent forces, to ensure the dimensional accuracy of your solutions.