Let's solve the **step by step: --- ##Step 1 Understanding the Problem** We given a planar and a partially filled table. We need to fill in: - \( n \): Number of links (including the frame/ground) - \( j \): Number of joints (**already given:** 12) - \( M \): Mobility (DOF, degrees of freedom) - **Idle DOF** --- ## **Step 2: Count the Links (\( n \))** Count all rigid bodies including the ground. Let's number them: 1. **Ground** 2. **Link attached to P** 3. **Link attached to S** 4. **Link attached between S and next joint** 5. **Link attached to middle slider** 6. **Link attached to rightmost slider** 7. **Link between two upper pivots (connecting all three sliders)** 8. **Link attached to R** After examining, there are **8 links** (including the ground). So, - \( n = 8 \) --- ## **Step 3: Number of Joints (\( j \))** Already **given as 12**. --- ## **Step 4: Mobility Formula (Gruebler’s Equation for Planar Mechanisms)** \[ M = 3(n-1) - 2j \] Where: - \( n = 8 \) - \( j = 12 \) Plug in values: \[ M = 3(8-1) - 2(12) \] \[ M = 3 \times 7 - 24 = 21 - 24 = -3 \] --- ## **Step 5: Idle DOF** **Idle DOF** is the number of degrees of freedom that are not used for the intended motion, often due to redundant constraints. If \( M \) is negative, it means the mechanism is over-constrained, and the absolute value of \( M \) is the number of idle DOF. \[ \text{idle DOF} = |M| = 3 \] --- ## **Step 6: Fill in the Table** | n | j | M | idle DOF | |----|----|----|----------| | 8 | 12 | -3 | 3 | --- ## **Final Answers** - **n (Number of links):** 8 - **j (Number of joints):** 12 - **M (Mobility):** -3 - **Idle DOF:** 3 --- ### **Summary Table** | n | j | M | idle DOF | |----|----|----|----------| | 8 | 12 | -3 | 3 | Let me know if you want step-by-step on how to count the links and joints!