Let's break down the problem step by step. --- ## **Step 1: Forces Acting on the Rod** Let's identify **all the forces** acting on the rod: 1. **Weight of the rod (W = 20 N)**, acting downward at the center. 2. **Normal reaction at the floor (N)**, acting upward at the right end. 3. **Frictional force at the floor (f)**, acting horizontally at the right end (opposing slipping). 4. **Reaction at the hinge (H)**, which can be resolved into horizontal (Hₓ) and vertical (Hᵧ) components at the left end. - **Total number of forces acting:** 1. Weight (W) 2. Normal reaction (N) 3. Friction (f) 4. Hinge reaction (H) (but it's a single reaction force, even if it has components) > **Note:** The hinge reaction, despite being resolved into two components, is considered **one force** (since it's a single contact point and a single constraint). > > **Total forces:** **4** --- ## **Step 2: Die Throw Condition** - The die is fair and six-sided, so possible outcomes: {1, 2, 3, 4, 5, 6} - We need the number on the die to be **less than or equal to the number of forces acting on the rod** (**≤ 4**). - Favorable outcomes: {1, 2, 3, 4} - Total possible outcomes: 6 --- ## **Step 3: Probability Calculation** \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{6} = \frac{2}{3} \] --- ## **Step 4: Final Answer** \[ \boxed{\frac{2}{3}} \] --- ### **Summary Table** | Step | Explanation | Value | |-----------------------------|------------------------------------------|----------------| | Number of forces on the rod | Weight, normal, friction, hinge reaction | 4 | | Favorable die outcomes | 1, 2, 3, 4 | 4 | | Total die outcomes | | 6 | | Probability | Favorable/Total | \(\frac{2}{3}\) | --- **Final Answer:** The probability is \(\boxed{\frac{2}{3}}\).