Question 2.SP.5: Determine the magnitude and direction of the smallest force ......
Determine the magnitude and direction of the smallest force F that maintains the 30-kg package shown in equilibrium. Note that the force exerted by the rollers on the package is perpendicular to the incline.
STRATEGY: This is an equilibrium problem with three coplanar forces that you can solve with a force triangle. The new wrinkle is to determine a minimum force. You can approach this part of the solution in a way similar to Sample Problem 2.2.
- Mass of the package: 30 kg
- The force exerted by the rollers on the package is perpendicular to the incline.
Final Answer
MODELING and ANALYSIS:
Free-Body Diagram. Choose the package as a free body, assuming that it can be treated as a particle. Then draw the corresponding free-body diagram (Fig. 1).
Equilibrium Condition. Since only three forces act on the free body, draw a force triangle to express that it is in equilibrium (Fig. 2). Line 1-1′ represents the known direction of P. In order to obtain the minimum value of the force F, choose the direction of F to be perpendicular to that of P. From the geometry of this triangle,
F=(294 \mathrm{~N}) \sin 15^{\circ}=76.1 \mathrm{~N} \quad \alpha=15^{\circ}
\mathbf{F}=76.1 \mathrm{~N} \measuredangle 15^{\circ}
REFLECT and THINK: Determining maximum and minimum forces to maintain equilibrium is a common practical problem. Here the force needed is about 25% of the weight of the package, which seems reasonable for an incline of 15°.
