## Step 1: Analyze Forces and Friction for Each Package ### Forces on Each Package Each package has: - Weight \( mg \) acting downward. - Normal force \( N \) perpendicular to the belt. - Friction force \( f \) acting up the incline (resisting motion down the incline). #### Calculate Weight Components for Each Package (L, M, N) - Mass, \( m = 72.\, \text{kg} \) - Gravity, \( g = 9.81\, \text{m/s}^2 \) - Incline angle, \( \theta = 15^\circ \) Weight component **down the incline**: \[ F_{\text{down}} = mg \sin \theta = 72. \times 9.81 \times \sin(15^\circ) \approx 183.2\, \text{N} \] Weight component **perpendicular to incline**: \[ N = mg \cos \theta = 72. \times 9.81 \times \cos(15^\circ) \approx 682.6\, \text{N} \] **Explanation:** Each package tries to slide down due to the component of its weight parallel to the incline; friction resists this. --- ## Step 2: Find Maximum Static Friction and Compare ### Calculate Maximum Static Friction for Each Package - For **L and N**: \( \mu_s = .30 \) - For **M**: \( \mu_s = .10 \) **Maximum static friction for L/N:** \[ f_{s,\text{max,L/N}} = \mu_s N = .30 \times 682.6 \approx 204.8\, \text{N} \] **Maximum static friction for M:** \[ f_{s,\text{max,M}} = .10 \times 682.6 \approx 68.3\, \text{N} \] **Explanation:** The friction force must be at least as large as the downhill force to prevent motion. If not, the package will move. --- ### **Check If Any Package Moves** - **Force down the incline for each package:** \( 183.2\, \text{N} \) - **L/N max friction:** \( 204.8\, \text{N} \) (enough) - **M max friction:** \( 68.3\, \text{N} \) (**NOT enough**) **Package M will move** because the required force (183.2 N) exceeds the max static friction (68.3 N). --- ## **Final Answer** **The required frictional force acting on package L to maintain equilibrium:** \[ f_{s,\text{L,required}} = F_{\text{down}} = 183.2\, \text{N} \] However, **package M will move** since its friction is insufficient, so the system cannot remain in equilibrium as specified. **If only considering L's frictional force (before slipping):** \[ \boxed{183\, \text{N}} \] But, the system will not stay at rest because package M will slip.