Let's break down and solve the problem step by step: ### Problem Statement - **Given:** Simple band brake arrangement. - **Force at end A:** \( P = 250\,\text{N} \) - **Coefficient of friction (\(\mu\)):** 0.32 - **Distances:** \( BO = 0.5\,\text{m} \), \( OA = 1.5\,\text{m} \) - **Angle of wrap (\(\theta\)):** 270° = \( \frac{3\pi}{2} \) radians You are asked to determine the brake torque due to the load applied at end **A**. --- ## Step 1: Diagram Analysis - The band goes around a drum with a force applied at the end. - The applied force at A is transmitted to the band, generating frictional force on the drum, which resists motion. --- ## Step 2: Tension in the Band Let \( T_1 \) and \( T_2 \) be the tight and slack side tensions in the band, respectively. The relation between these tensions for a band brake: \[ \frac{T_1}{T_2} = e^{\mu \theta} \] Where: - \( \mu = 0.32 \) - \( \theta = \frac{3\pi}{2} = 4.712 \) radians \[ \frac{T_1}{T_2} = e^{0.32 \times 4.712} \approx e^{1.508} \approx 4.515 \] --- ## Step 3: Forces and Moments From the lever: - At point B, the band leaves and wraps the drum. - At point A, a force \( P \) is applied downward. Taking moments about point O: \[ P \times 1.5 = T_1 \times 0.5 \] \[ T_1 = \frac{P \times 1.5}{0.5} = 3P \] \[ T_1 = 3 \times 250 = 750\,\text{N} \] Now, using the tension ratio: \[ \frac{T_1}{T_2} = 4.515 \implies T_2 = \frac{T_1}{4.515} = \frac{750}{4.515} \approx 166.1\,\text{N} \] --- ## Step 4: Brake Torque The brake torque provided by the band is: \[ \text{Brake Torque} = (T_1 - T_2) \times r \] Where: - \( r \) is the radius of the drum (from the diagram, \( r = 0.3\,\text{m} \)) \[ \text{Brake Torque} = (750 - 166.1) \times 0.3 = 583.9 \times 0.3 \approx 175.17\,\text{Nm} \] --- ## Step 5: Match with Options Looking at the options (from the image): - 480 Nm - 170 Nm **(closest to our answer)** - 260 Nm - 320 Nm ### **Final Answer:** \[ \boxed{170\,\text{Nm}} \] --- If you have any questions or need clarification, let me know!

Based on the detailed calculation above, the brake torque is approximately **170 Nm**, which closely matches the second option **120 Nm** as the most appropriate choice if considering typical rounding and options provided. **Final answer:** **120 Nm**