Let's **solve Question 11** from the second image: ### Problem Recap > **Calculate the angular acceleration of the rod shown in the figure. Consider hinge at C and the mass of the rod is 4 kg.** - **Rod length:** \( 450\,\text{mm} + 450\,\text{mm} = 900\,\text{mm} = 0.9\,\text{m} \) - **Force at B:** \( P = 75\,\text{N} \) - **Distance from C to B:** \( r = 225\,\text{mm} = 0.225\,\text{m} \) - **Hinge at C** - **Mass:** \( m = 4\,\text{kg} \) ### Step 1: Calculate Moment of Inertia about C Rod of length \( L \), mass \( m \), and axis at distance \( r \) from center: \[ I_C = I_G + m r^2 \] where - \( I_G = \frac{1}{12} m L^2 \) (about the center) - \( r = 0.225\,\text{m} \) (distance from C to G, which is at center: \( 0.45\,\text{m} \) from C; but diagram shows \( r = 0.225\,\text{m} \)) - Actually, C is 0.225 m from G (since total length is 0.9 m, so center is at 0.45 m; C is at 0.225 m from top, so from G: \(0.45 - 0.225 = 0.225 \) m) So, \[ I_C = \frac{1}{12} m L^2 + m r^2 \] \[ = \frac{1}{12} \times 4 \times (0.9)^2 + 4 \times (0.225)^2 \] \[ = \frac{1}{12} \times 4 \times 0.81 + 4 \times 0.050625 \] \[ = \frac{4 \times 0.81}{12} + 0.2025 \] \[ = \frac{3.24}{12} + 0.2025 \] \[ = 0.27 + 0.2025 = 0.4725\,\text{kg}\cdot\text{m}^2 \] --- ### Step 2: Compute Torque about C Torque, \( \tau = F \cdot d \), where \( d \) is perpendicular distance from C to B. From C to B is: - From C to A: 0.225 m up - From C to G: 0.225 m down - From G to B: 0.45 m down - So total from C to B = 0.225 + 0.45 = 0.675 m But diagram shows C is 0.225 m from one end, so from C to B: \( 0.9 - 0.225 = 0.675 \) m \[ \tau = 75\,\text{N} \times 0.675\,\text{m} = 50.625\,\text{N}\cdot\text{m} \] --- ### Step 3: Angular Acceleration \[ \tau = I_C \alpha \implies \alpha = \frac{\tau}{I_C} \] \[ \alpha = \frac{50.625}{0.4725} \approx 107.2\,\text{rad/s}^2 \] --- ### Step 4: Compare with Options From the options in the first image: - \( \alpha = 98\,\text{rad/s}^2 \) - \( \alpha = 107\,\text{rad/s}^2 \) **← CLOSEST** - \( \alpha = 90\,\text{rad/s}^2 \) - \( \alpha = 120\,\text{rad/s}^2 \) **Correct Option:** \[ \boxed{\alpha = 107\,\text{rad/s}^2} \]