To solve these problems, we need to analyze each mechanism and determine the number of degrees of freedom (DoF) using the Grler’s Equation for planar mechanisms: \text{DoF} 3(n-1) -2j - h \] : - \( n \) = number of (including the). - \( \) = number of lower (full joints, like revolute or prismatic pairs). - \( h \) = number of higher pairs (half joints, such as cam or gear contacts). Let's break down each mechanism step-by-step: ### a) Analysis 1. **Count the Links:** - There are 5 links including the ground. 2. **Count the Joints:** - There are 6 revolute joints. 3. **Apply Gruebler’s Equation:** \[ \text{DoF} = 3(5-1) - 2(6) = 12 - 12 = \] ### b) Analysis 1. **Count the Links:** - There are 6 links including the ground. 2. **Count the Joints:** - There are 7 revolute joints. 3. **Apply Gruebler’s Equation:** \[ \text{DoF} = 3(6-1) - 2(7) = 15 - 14 = 1 \] ### c) Analysis 1. **Count the Links:** - There are 5 links including the ground. 2. **Count the Joints:** - There are 6 revolute joints. 3. **Apply Gruebler’s Equation:** \[ \text{DoF} = 3(5-1) - 2(6) = 12 - 12 = \] ### d) Analysis 1. **Count the Links:** - There are 4 links including the ground. 2. **Count the Joints:** - There are 4 revolute joints. 3. **Apply Gruebler’s Equation:** \[ \text{DoF} = 3(4-1) - 2(4) = 9 - 8 = 1 \] ### e) Analysis 1. **Count the Links:** - There are 6 links including the ground. 2. **Count the Joints:** - There are 7 revolute joints. 3. **Apply Gruebler’s Equation:** \[ \text{DoF} = 3(6-1) - 2(7) = 15 - 14 = 1 \] ### f) Analysis 1. **Count the Links:** - There are 6 links including the ground. 2. **Count the Joints:** - There are 8 revolute joints. 3. **Apply Gruebler’s Equation:** \[ \text{DoF} = 3(6-1) - 2(8) = 15 - 16 = -1 \] ### Summary of Degrees of Freedom - **a)** DoF = - **b)** DoF = 1 - **c)** DoF = - **d)** DoF = 1 - **e)** DoF = 1 - **f)** DoF = -1 Negative DoF indicates an over-constrained mechanism.