Let's solve the problem **step-by-step**: --- ### **Step 1: Given Data** - **Mass of OA (m₁):** 2.9 kg - **Mass of Gear D (m₂):** 5.9 kg - **Radius of gyration of D (k):** 127 mm = 0.127 m - **OA length:** 380 mm = 0.38 m - **Distance OA to center of D:** 175 mm = 0.175 m - **Gravitational acceleration (g):** 9.81 m/s² --- ### **Step 2: Moments of Inertia** #### **Arm OA (about O):** Uniform rod about one end: \[ I_{OA} = \frac{1}{3} m_1 L^2 = \frac{1}{3} \times 2.9 \times (0.38)^2 = \frac{1}{3} \times 2.9 \times 0.1444 = 0.1397 \ \text{kg}\cdot\text{m}^2 \] #### **Gear D (about its center):** \[ I_{D,\text{center}} = m_2 k^2 = 5.9 \times (0.127)^2 = 5.9 \times 0.016129 = 0.0951 \ \text{kg}\cdot\text{m}^2 \] But D's center is a distance 0.555 m (0.38 + 0.175) from O. **Parallel Axis Theorem:** \[ I_{D,\text{O}} = I_{D,\text{center}} + m_2 d^2 = 0.0951 + 5.9 \times (0.555)^2 = 0.0951 + 5.9 \times 0.3080 = 0.0951 + 1.8172 = 1.9123 \ \text{kg}\cdot\text{m}^2 \] --- ### **Step 3: Total Moment of Inertia about O** \[ I_{\text{total}} = I_{OA} + I_{D,\text{O}} = 0.1397 + 1.9123 = 2.052 \ \text{kg}\cdot\text{m}^2 \] --- ### **Step 4: Torques due to Gravity** #### **Torque due to OA:** \[ \tau_{OA} = m_1 g \left(\frac{L}{2}\right) = 2.9 \times 9.81 \times \frac{0.38}{2} = 2.9 \times 9.81 \times 0.19 = 5.397 \ \text{N}\cdot\text{m} \] #### **Torque due to D:** \[ \tau_{D} = m_2 g r_D = 5.9 \times 9.81 \times 0.555 = 5.9 \times 5.44255 = 32.109 \ \text{N}\cdot\text{m} \] --- ### **Step 5: Net Torque** \[ \tau_{\text{net}} = \tau_{OA} + \tau_{D} = 5.397 + 32.109 = 37.506 \ \text{N}\cdot\text{m} \] (Torques are both clockwise, so negative direction if CCW is positive.) --- ### **Step 6: Relating Torque and Angular Acceleration** \[ \tau = I \alpha \] \[ \alpha = \frac{\tau_{\text{net}}}{I_{\text{total}}} = \frac{37.506}{2.052} = 18.28 \ \text{rad/s}^2 \] Since the torque is clockwise, the angular acceleration is **negative**. --- ### **Step 7: Final Answer** \[ \boxed{\alpha = -18.3 \ \text{rad/s}^2} \] --- **Explanation:** - Calculated the moments of inertia for the arm and gear. - Summed the torques due to gravity. - Used Newton's second law for rotation to find the angular acceleration. - The negative sign indicates **clockwise** acceleration. --- Let me know if you need a more detailed breakdown or clarification!