Let's break down the problem step by step. --- ## **Step 1: Understand the Problem** - **Given:** - Crank length (\( l_2 \)) = 12 units (with a scale of 1:10, real length = 120 units) - Crank rotates at \( N = 100 \) rpm (revolutions per minute) - Need to find: **velocity** (\( v_M \)), **acceleration** (\( a_M \)), and **radius of curvature** (\( \rho_M \)) of point M. --- ## **Step 2: Convert Units** - **Angular velocity** (\( \omega \)): \[ \omega = 2\pi N / 60 \] \[ \omega = 2\pi \times 100 / 60 \approx 104.72\, \text{rad/s} \] - **Crank Length**: \( l_2 = 12 \times 10 = 120 \) units --- ## **Step 3: Calculate Linear Velocity of the Crank End** \[ v_{A} = \omega \cdot r = 104.72 \times 120 = 12,566.4 \ \text{units/s} \] --- ## **Step 4: Kinematic Invariants for Point M** For a coupler point M on a slider-crank, assuming it's at the end of the crank (worst-case, if no extra dimensions are given): - **Velocity of M** (instantaneous): \[ v_M = v_A = 12,566.4 \ \text{units/s} \] (If M is not colinear with A, more info would be needed.) - **Acceleration of M** (centripetal, as the crank rotates at a constant speed): \[ a_M = \omega^2 \cdot r = (104.72)^2 \times 120 = 1,317,504 \ \text{units/s}^2 \] --- ## **Step 5: Radius of Curvature** \[ \rho_M = \frac{v_M^2}{a_M} \] \[ \rho_M = \frac{(12,566.4)^2}{1,317,504} \approx 120 \ \text{units} \] --- ## **Step 6: Final Answers** ### **Summary Table** | Quantity | Formula | Value | |------------------|-----------------------|-----------------| | Angular velocity (\(\omega\)) | \(2\pi N / 60\) | \(104.72\, \text{rad/s}\) | | Velocity (\(v_M\)) | \(\omega \cdot r\) | \(12,566.4\, \text{units/s}\) | | Acceleration (\(a_M\)) | \(\omega^2 \cdot r\) | \(1,317,504\, \text{units/s}^2\) | | Radius of curvature (\(\rho_M\)) | \(v_M^2/a_M\) | \(120\, \text{units}\) | --- ### **Final Answers** - **Velocity of M**: \(12,566.4\, \text{units/s}\) - **Acceleration of M**: \(1,317,504\, \text{units/s}^2\) - **Radius of curvature at M**: \(120\, \text{units}\) --- #### **Assumptions** - M is at the tip of the crank (maximal values). - All units are consistent (for "units", substitute the actual length unit if specified). If you have more geometric details about M's position, you can refine these results!