Let's break the solution into **two steps** for each subquestion (a, b, and c), with explanations and final answers. --- ## a) Determine the Maximum Actuating Force ### **Step 1: Calculate the Maximum Force per Shoe** - **Given:** - Diameter of drum, \(D 400 \text{}\) ⇒ Radius, \(r = 200 \text{ mm} = .2 \text{ m}\) - Face width of shoe, \(b = 75 \text{ mm} = .075 \text{ m}\) - Maximum pressure, \(P_{max} = 100 \text{ kPa} = 1,000,000 \text{ N/m}^2\) - **Contact area per shoe:** - Each shoe covers \(60^\circ\) (from diagram, \(15^\circ + 30^\circ + 15^\circ\)) = \(\frac{60}{360} = \frac{1}{6}\) of the circumference. - Arc length per shoe = \( \frac{1}{6} \times 2\pi \times .2 = \frac{2\pi \times .2}{6} \approx .209 \text{ m} \) - Contact area per shoe = arc length × width = \(.209 \times .075 \approx .0157 \text{ m}^2\) - **Maximum force per shoe:** \[ F_{max} = P_{max} \times \text{Area} = 1,000,000 \times .0157 = 15,700 \text{ N} \] **Explanation:** We calculated the area of contact for each shoe and used the maximum allowable pressure to find the maximum force that can be applied by each shoe. --- ### **Step 2: Calculate Maximum Actuating Force** - There are 4 shoes, but each pin (A or B) supports 2 shoes. - The actuating force is the maximum force required per pin. \[ \text{Maximum actuating force per pin} = 2 \times F_{max} = 2 \times 15,700 = 31,400 \text{ N} \] **Explanation:** Since each actuating pin must support two shoes, we multiply the maximum force per shoe by 2. --- ### **Final Answer (a):** \[ \boxed{31,400 \text{ N}} \] is the maximum actuating force per pin. --- ## b) Estimate the Brake Capacity ### **Step 1: Calculate the Total Friction Force** - Coefficient of friction, \(\mu = .24\) - Each shoe friction force = \(\mu \times F_{max} = .24 \times 15,700 = 3,768 \text{ N}\) - Total friction force (all 4 shoes) = \(4 \times 3,768 = 15,072 \text{ N}\) **Explanation:** Friction force per shoe is the product of normal force and coefficient of friction, summed for all shoes. --- ### **Step 2: Calculate the Brake Torque (Capacity)** - Effective radius = \(.2 \text{ m}\) - Total torque (brake capacity): \[ \text{Brake torque} = \text{Total friction force} \times \text{radius} = 15,072 \times .2 = 3,014.4 \text{ Nm} \] **Explanation:** Brake torque is the product of total friction force and the drum radius. --- ### **Final Answer (b):** \[ \boxed{3,014.4 \text{ Nm}} \] is the brake capacity. --- ## c) Estimate the Hinge-Pin Reactions ### **Step 1: Find the Force Components at Each Pin** - Each pin supports two shoes, each with normal force \(F_{max}\). - The direction of these forces is symmetrically spaced (see diagram). - For pin A (similarly for B): - The angle between the two forces is \(30^\circ\) (from diagram). - The resultant force at the pin: \[ R = 2F_{max}\cos\left(\frac{30^\circ}{2}\right) = 2 \times 15,700 \times \cos(15^\circ) \] \[ = 31,400 \times .9659 = 30,320 \text{ N} \] **Explanation:** We use vector addition since the two forces are at an angle to each other. --- ### **Step 2: Find the Direction and Report Result** - The resultant acts along the angle bisector between the two shoe forces. - The reaction at each pin = \(30,320 \text{ N}\). **Explanation:** Both pins have the same reaction due to symmetry. --- ### **Final Answer (c):** \[ \boxed{30,320 \text{ N}} \] reaction at each hinge pin A and B.