Let's break down the given problem and work through the solution step by step. --- ### **Given Data:** - Crank length (\(r\)) = **50 mm** - Connecting rod length (\(l\)) = **200 mm** - Crank angle (\(\theta\)) = **65°** - Crank speed (\(N\)) = **300 rpm** (clockwise) --- ### **Step 1: Find Angular Velocity (\(\omega\))** \[ \omega = 2\pi N / 60 \] \[ \omega = 2\pi \times 300 / 60 = 31.42 \text{ rad/s} \] --- ### **Step 2: Find Angular Acceleration (\(\alpha\))** Since the speed is constant, \[ \alpha = 0 \] --- ### **Step 3: Find Linear Acceleration of the Piston (\(a_P\))** #### **Standard Acceleration Analysis** The acceleration of the piston can be calculated as: \[ a_P = -r\omega^2 \cos\theta - \frac{r^2\omega^2}{l} \cos(2\theta) \] #### **Plug in the Values** - \(r = 0.05\) m - \(\omega = 31.42\) rad/s - \(\theta = 65^\circ\) - \(l = 0.2\) m --- \[ a_P = -0.05 \times (31.42)^2 \times \cos(65^\circ) - \frac{(0.05)^2 \times (31.42)^2}{0.2} \cos(130^\circ) \] Calculate each part: - \((31.42)^2 = 987.1\) - \(\cos(65^\circ) \approx 0.4226\) - \(\cos(130^\circ) \approx -0.6428\) First term: \[ -0.05 \times 987.1 \times 0.4226 \approx -20.86 \] Second term: \[ (0.05)^2 = 0.0025 \] \[ 0.0025 \times 987.1 = 2.46775 \] \[ 2.46775 / 0.2 = 12.33875 \] \[ 12.33875 \times (-0.6428) = -7.937 \] Total acceleration: \[ a_P = -20.86 - (-7.937) = -20.86 + 7.937 = -12.92\, \text{m/s}^2 \] --- ### **Final Answer** \[ \boxed{a_P \approx -12.9\, \text{m/s}^2} \] --- #### **Graphical Method** If you need to solve this graphically: 1. **Draw the crank and connecting rod to scale** at the given angle. 2. **Draw acceleration polygons** (Klein’s Construction or similar). 3. Use the scale to measure the acceleration of the piston. But for quick calculation, the analytical method above gives you the answer. --- If you need a step-by-step graphical solution (with construction), let me know!