Let's solve the problem step-by-step for each planar linkage using **Gruebler’s formula**. --- ## **Gruebler’s Formula (for planar mechanisms):** \[ M = 3(n - 1) - 2j_1 - j_2 \] Where: - \( M \) = Mobility (degrees of freedom) - \( n \) = Number of links (including frame) - \( j_1 \) = Number of lower pairs (single DOF joints: revolute/prismatic) - \( j_2 \) = Number of higher pairs (two DOF joints: cams, gears, etc.) For most simple planar mechanisms, \( j_2 = \). --- # (a) **First Linkage** ### **Step 1: Count number of links (\( n \))** - 2 moving links + 1 fixed frame = **3 links** ### **Step 2: Count number of lower pairs (\( j_1 \))** - 2 revolute joints (pins at ground) = **2** - 1 revolute joint between the two links = **1** - **Total \( j_1 = 3 \)** ### **Step 3: Plug into Gruebler’s Formula** \[ M = 3(3-1) - 2 \times 3 = 6 - 6 = \] ### **Final Answer (a):** - Number of links: **3** - Number of joints: **3** - Mobility: **** (Structure, not a mechanism) --- # (b) **Second Linkage (Slider-Crank)** ### **Step 1: Count number of links (\( n \))** - 2 rods (crank and coupler) + 1 slider + 1 frame = **4 links** ### **Step 2: Count number of lower pairs (\( j_1 \))** - 2 revolute joints (at ground and between links) = **2** - 1 revolute joint (between coupler and slider) = **1** - 1 prismatic joint (slider) = **1** - **Total \( j_1 = 4 \)** ### **Step 3: Plug into Gruebler’s Formula** \[ M = 3(4-1) - 2 \times 4 = 9 - 8 = 1 \] ### **Final Answer (b):** - Number of links: **4** - Number of joints: **4** - Mobility: **1** (One input controls the mechanism) --- # (c) **Third Linkage (Watt or Stephenson Six-Bar)** ### **Step 1: Count number of links (\( n \))** - 6 moving links + 1 frame = **7 links** ### **Step 2: Count number of lower pairs (\( j_1 \))** - Count the joints: Each dot is a revolute joint. - There are **9** joints. ### **Step 3: Plug into Gruebler’s Formula** \[ M = 3(7-1) - 2 \times 9 = 18 - 18 = \] ### **Final Answer (c):** - Number of links: **7** - Number of joints: **9** - Mobility: **** (It’s a structure—not a mechanism—unless one joint is ground and one is input) --- # (d) **Fourth Linkage (Double Rocker with Sliders)** ### **Step 1: Count number of links (\( n \))** - 2 rods + 2 sliders + 1 frame = **5 links** ### **Step 2: Count number of lower pairs (\( j_1 \))** - 2 revolute joints at ground = **2** - 2 prismatic joints (sliders) = **2** - 2 revolute joints between rods and sliders = **2** - **Total \( j_1 = 6 \)** ### **Step 3: Plug into Gruebler’s Formula** \[ M = 3(5-1) - 2 \times 6 = 12 - 12 = \] ### **Final Answer (d):** - Number of links: **5** - Number of joints: **6** - Mobility: **** (Structure, not a mechanism) --- ## **Summary Table** | Linkage | Links (\( n \)) | Joints (\( j_1 \)) | Mobility (\( M \)) | |:-------:|:---------------:|:------------------:|:------------------:| | (a) | 3 | 3 | | | (b) | 4 | 4 | 1 | | (c) | 7 | 9 | | | (d) | 5 | 6 | | --- **Let me know if you need explanations for each mechanism!**