Let's solve the problem step by step. --- ## **Step 1: Understanding the System** This is a basic **pulley system** (movable pulley). - The **bucket** has a weight \( W = 100 \) lb. - You apply a **force \( F \)** downward at point \( A \). - The **cord passes over a fixed pulley**, then through a movable pulley attached to the bucket. --- ## **Step 2: Relating Cord Lengths** Let: - \( s_A \) = distance from the support to point A (where the force is applied) - \( s_B \) = distance from the support to the bucket Since the total length \( l \) of the cord is constant, we have: \[ s_A + 2s_B = \text{constant} \] - When you pull the cord at \( A \) down by a distance \(\Delta s_A\), the bucket rises by \(\Delta s_B\), such that \[ \Delta s_A + 2\Delta s_B = \implies \Delta s_A = -2\Delta s_B \] - Thus, **for every 2 units the cord is pulled down at \( A \), the bucket rises by 1 unit.** --- ## **Step 3: Virtual Work Principle** Let’s use **virtual work** to relate the forces. - The **work done by the applied force \( F \)** is: \[ \delta W_F = F\,\delta s_A \] - The **work done against the weight \( W \)** is: \[ \delta W_W = -W\,\delta s_B \] (Negative because the weight acts downward, but the bucket moves up.) **Set the total virtual work to zero:** \[ F\,\delta s_A - W\,\delta s_B = \] From above: \( \delta s_A = -2\,\delta s_B \) Plug in: \[ F \cdot (-2\,\delta s_B) - W\,\delta s_B = \\ (-2F - W)\delta s_B = \] Since \(\delta s_B \neq \): \[ -2F - W = \implies 2F = -W \implies F = -\frac{W}{2} \] The negative means the force is in the opposite direction to \( W \) (which matches the diagram; you pull down to lift the bucket). So, **magnitude**: \[ F = \frac{W}{2} \] --- ## **Step 4: Plug in the Values** Given \( W = 100 \) lb: \[ F = \frac{100}{2} = 50~\text{lb} \] --- ## **Final Answer** ### **The force \( F \) needed to lift the bucket is:** \[ \boxed{50~\text{lb}} \] --- ### **Summary Table** | Parameter | Value | |-----------|----------------| | Weight \( W \) | 100 lb | | Required force \( F \) | 50 lb | --- **Key Concept:** *A movable pulley system with one movable and one fixed pulley gives a mechanical advantage of 2. Thus, the force required is half the weight.*