# Step-by-Step Solution Let's break down the problem and find the **vector loop equation** for the given four-bar mechanism. --- ## **Step 1: Understanding the Mechanism** - **Link 2** (Crank): The input link rotating about a fixed point. - **Link 3** (Coupler): Connects link 2 and link 4. - **Link 4** (Slider): Moves linearly. - **Ground (Frame)**: Fixed points. --- ## **Step 2: Assign Notation and Vectors** Let’s assign: - \( r_2 \): Length of link 2 (crank) - \( r_3 \): Length of link 3 (coupler) - \( r_4 \): Length of link 4 (slider, variable distance) - \( \theta_2 \): Angle of link 2 (input, **known**) - \( \theta_3 \): Angle of link 3 (unknown) - \( \theta_4 \): Angle of link 4 (slider, usually or π, as it's horizontal) --- ## **Step 3: Draw the Vector Loop** The vector loop for the mechanism is: \[ \vec{r}_2 + \vec{r}_3 = \vec{r}_4 \] But in proper closing form (starting and ending at the fixed point): \[ \vec{r}_2 + \vec{r}_3 = \vec{r}_4 \] So, more precisely: \[ \vec{r}_2 + \vec{r}_3 - \vec{r}_4 = \] ### **Express Each Vector in Components** Assuming the ground (frame) is on the x-axis: - \( \vec{r}_2 = r_2 \cos\theta_2 \hat{i} + r_2 \sin\theta_2 \hat{j} \) - \( \vec{r}_3 = r_3 \cos\theta_3 \hat{i} + r_3 \sin\theta_3 \hat{j} \) - \( \vec{r}_4 = r_4 \hat{i} \) (slider moves along x-direction, so angle is ) --- ## **Step 4: Write the Vector Loop Equation in Components** \[ (r_2 \cos\theta_2 + r_3 \cos\theta_3 - r_4)\hat{i} + (r_2 \sin\theta_2 + r_3 \sin\theta_3)\hat{j} = \] This gives two scalar equations (since vectors must sum to zero): ### **Constraint Equations** 1. **x-component:** \[ r_2 \cos\theta_2 + r_3 \cos\theta_3 - r_4 = \] 2. **y-component:** \[ r_2 \sin\theta_2 + r_3 \sin\theta_3 = \] --- ## **Step 5: Identify Constants, Variables, and Constraints** - **Knowns (Constants):** - \( r_2 \): Length of link 2 (input crank) - \( r_3 \): Length of link 3 (coupler) - \( \theta_2 \): Input angle (since link 2 is input) - **Unknowns (Variables):** - \( r_4 \): Position of slider (output) - \( \theta_3 \): Angle of coupler - **Constraints (Equations):** - \( r_2 \cos\theta_2 + r_3 \cos\theta_3 - r_4 = \) - \( r_2 \sin\theta_2 + r_3 \sin\theta_3 = \) --- ## **Step 6: Final Answers** ### **a) Vector Loop Equation** \[ \vec{r}_2 + \vec{r}_3 - \vec{r}_4 = \] ### **b) Scalar Constraint Equations** \[ \begin{cases} r_2 \cos\theta_2 + r_3 \cos\theta_3 - r_4 = \\ r_2 \sin\theta_2 + r_3 \sin\theta_3 = \end{cases} \] ### **c) Knowns, Unknowns, Constraints** - **Knowns:** \( r_2, r_3, \theta_2 \) - **Unknowns:** \( \theta_3, r_4 \) - **Constraints:** Two equations above --- ## **Summary Table** | Quantity | Type | Description | |-----------------|-----------|------------------------------------| | \( r_2 \) | Known | Length of input crank | | \( r_3 \) | Known | Length of coupler | | \( \theta_2 \) | Known | Input angle | | \( \theta_3 \) | Unknown | Coupler angle | | \( r_4 \) | Unknown | Slider position (output) | | Constraint 1 | Equation | \( r_2\cos\theta_2 + r_3\cos\theta_3 - r_4 = \) | | Constraint 2 | Equation | \( r_2\sin\theta_2 + r_3\sin\theta_3 = \) | --- **If you need a numerical solution or further clarification, just ask!**