## Determining the Mobility of the Linkage Let's break down the problem step by step! ### Step 1: **Identify the Type of Linkage** The diagram shows a planar mechanism with three pistons connected to a common crank through connecting rods. This is similar to a **planar slider-crank mechanism**. ### Step 2: **Count the Links (n)** - **Ground link** (the body/frame): 1 - **Crank** (center rotating arm): 1 - **Connecting rods** (3 rods): 3 - **Sliders** (pistons): 3 **Total links, n = 1 (ground) + 1 (crank) + 3 (rods) + 3 (sliders) = 8** ### Step 3: **Count the Joints (j)** #### Types of joints: - **Revolute (pin) joints** (crank to rods, crank to ground): - Crank to each rod = 3 - Crank to ground (center) = 1 - Each rod to each piston = 3 - **Total revolute joints = 3 (crank-rod) + 1 (crank-ground) + 3 (rod-piston) = 7** - **Prismatic (sliding) joints** (pistons in cylinders): 3 **Total joints, j = 7 (revolute) + 3 (prismatic) = 10** ### Step 4: **Use the Mobility Equation (Gruebler’s Equation for Planar Mechanisms)** \[ M = 3(n-1) - 2j \] Where: - \( M \) = mobility (degrees of freedom) - \( n \) = number of links - \( j \) = number of lower pairs (joints) Plug in the numbers: \[ M = 3(8-1) - 2(10) = 3(7) - 20 = 21 - 20 = 1 \] ### **Final Answer** \[ \boxed{1} \] #### **Explanation** - The mobility is **1**, which means the entire mechanism only has **one independent motion** (for example, turning the central crank will move all the pistons in a fixed relationship). --- **Let me know if you want a deeper breakdown of how I counted the links or joints!**