Let's go through each part of this problem.

1. Problem Summary:
   • A force $F(t) = \beta t^2 + \gamma$ is applied to a rotating rigid body with rotational inertia $I_0$, at a fixed moment arm $R$ from the axis of rotation.
   • We need to derive expressions for the angular impulse, determine the angular momentum after a specific time, and calculate quantities like angular speed, translational speed, and instantaneous power given certain initial conditions.

Part (a): Deriving Angular Impulse

• Angular Impulse $J_{\text{rot}}$ over a time interval $t$ is given by: $J_{\text{rot}} = \int_0^t \tau(t) \, dt$ where $\tau(t)$ is the torque applied by the force.
• Torque $\tau(t)$ can be calculated as: $\tau(t) = R \cdot F(t) = R (\beta t^2 + \gamma)$
• Substituting $\tau(t)$ into the integral: $J_{\text{rot}} = \int_0^t R (\beta t^2 + \gamma) \, dt$
• Solving this integral: $J_{\text{rot}} = R \left( \int_0^t \beta t^2 \, dt + \int_0^t \gamma \, dt \right)$ After evaluating, this results in: $J_{\text{rot}} = R \left( \frac{\beta t^3}{3} + \gamma t \right)$

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

• Angular momentum $L$ can be found using the relationship $L = J_{\text{rot}}$ (if the object starts from rest).
• Substitute $t = 2$ into the expression derived in Part (a): $L = R \left( \frac{\beta (2)^3}{3} + \gamma \cdot 2 \right)$

Part (c): Calculations at $t = 3$ seconds

1. Angular Speed:

   • Given $L = I \omega$, solve for $\omega$: $\omega = \frac{L}{I}$
   • Use $J_{\text{rot}}$ at $t = 3$ for $L$.

2. Translational Speed at Location of Applied Force:

   • Translational speed $v = R \cdot \omega$.

3. Instantaneous Power after $t = 3$ seconds:

   • Power $P = \tau \cdot \omega$.
   • Substitute $\tau(t) = R(\beta t^2 + \gamma)$ and $\omega$ from previous calculations.

Would you like me to proceed with detailed calculations for each part?

Here are 5 questions for deeper understanding:

1. How is angular impulse related to torque in rotational dynamics?
2. Why is the moment arm important when calculating torque?
3. How does starting from rest affect angular momentum calculations?
4. What are the steps to compute instantaneous power in rotational systems?
5. How does rotational inertia impact angular speed?

Tip: Always keep track of units, especially with time-dependent forces and torques, to ensure consistent results.