## Step 1: Draw the Free Body Diagram Let’s focus on the **forces acting on the wedge**: - **F**: The vertical force applied downward. - **Normal force from the ground**: Acts vertically upward (let’s call it \( N \)). - **Reaction force from the applied load**: The wedge, due to the force F applied vertically, will experience a reaction force at the contact point that must be balanced by the ground and any constraint. Since there is **no friction** and the force application point does not move relative to the floor, the force F is transmitted directly downward at the point of contact. However, due to the wedge’s geometry, the reaction at the sloped surface (invisible in the diagram but implied by the force application) will have both vertical and horizontal components. **Key Point**: The **horizontal component** of the reaction force at the ground is what pushes the wedge to the left. This is the force we need to find as a function of \( F \). --- ## Step 2: Resolve Forces Using Geometry The force \( F \) applied vertically acts through the inclined plane at an angle \( \theta \). The **reaction force** at the interface can be broken into two components: - **Parallel to the base** (horizontal, leftward): \( F_{\text{left}} \) - **Perpendicular to the base** (vertical, upward): \( F_{\text{up}} \) Since the force is transmitted through the inclined plane, the **horizontal component** (pushing the wedge to the left) is: \[ F_{\text{left}} = F \cdot \cot(\theta) \] - **Explanation**: The vertical force F can be resolved into components parallel and perpendicular to the incline. The horizontal (leftward) component is found using trigonometry: - The force parallel to the base is \( F \cdot \cot(\theta) \) (since \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{\text{horizontal}}{\text{vertical}} \)). --- ## **Final Answer** \[ \boxed{F_{\text{left}} = F \cot(\theta)} \] - This is the horizontal force to the left on the wedge as a function of the initial vertical force \( F \) and the angle \( \theta \) of the wedge.