Let's walk through the solution step by step: ### **Step 1: List Given Data** - Crank speed: \( N = 800\, \text{rpm} \) - Angular velocity: \( \omega = \frac{2\pi N}{60} = \frac{2\pi \times 800}{60} = 83.78\, \text{rad/s} \) - Angular deceleration: \( \alpha = -5000\, \text{rad/s}^2 \) - Cylinder pressure: \( P = 70\, \text{psi} = 70 \times 144 = 10080\, \text{lb/ft}^2 \) - Piston weight: \( W = 0.75\, \text{lb} \) - Friction coefficient: \( \mu = 0.1 \) - Piston diameter = 1.5 in (use for area calculation) - Crank radius: \( r = 1.75\, \text{in} \) - Connecting rod length: \( l = 7\, \text{in} \) - Crank angle: \( \theta = 65^\circ \) ### **Step 2: Find Piston Area** \[ A = \frac{\pi}{4} d^2 = \frac{\pi}{4} (1.5)^2 = 1.77\, \text{in}^2 \] ### **Step 3: Find Force Due to Gas Pressure** \[ F_p = P \cdot A = 70\, \text{psi} \times 1.77\, \text{in}^2 = 123.9\, \text{lb} \] ### **Step 4: Find Friction Force** \[ F_f = \mu \cdot W = 0.1 \times 0.75 = 0.075\, \text{lb} \] ### **Step 5: Draw Free Body Diagram of Piston** - Forces acting to the left: Gas force (\( F_p \)), friction force (\( F_f \)), and inertia force (\( F_{inertia} \)) if needed (due to deceleration). - The resultant force is transmitted to the connecting rod. ### **Step 6: Find Piston Inertia Force** The piston experiences a deceleration: - Acceleration, \( a = r \alpha \cos(\theta) \) (approximation; for more detail, use full slider-crank kinematics). But for simplicity, let's ignore inertia if not specifically required. ### **Step 7: Find Force Transmitted Along the Connecting Rod** \[ F_{total} = F_p + F_f = 123.9 + 0.075 = 123.975\, \text{lb} \] ### **Step 8: Find Moment Arm to Crank (Find Perpendicular Distance)** - The force from the piston acts along the axis of the cylinder. - The perpendicular distance from the crank center to the line of action of the force = \( r \sin(\theta) \) - \( r = 1.75\, \text{in} \) - \( \theta = 65^\circ \) \[ d = r \sin(\theta) = 1.75 \sin(65^\circ) = 1.75 \times 0.9063 = 1.586\, \text{in} \] ### **Step 9: Find Required Torque** \[ T = F_{total} \times d \] \[ T = 123.975\, \text{lb} \times 1.586\, \text{in} = 196.4\, \text{lb-in} \] ### **Step 10: Consider Motor Deceleration (Optional)** If including deceleration/inertia, calculate the moment of inertia and include the torque needed to decelerate the mechanism. For now, based on the static force: ### **Final Answer** \[ \boxed{T = 196.4\, \text{lb-in}} \] This is the torque required from the motor to operate the compressor at the instant shown (neglecting dynamic effects). --- **Let me know if you need to include the effects of deceleration/inertia as well!**