A steplader of negligible weight is constructed as shown in Figure P 12.51 . A painter of mass 7

step 1 Here we have the pair of objects or the pair of the legs. We will draw the legs separately. So l

A steplader of negligible weight is constructed as shown in...

A steplader of negligible weight is constructed as shown in Figure $\mathrm{P} 12.51 .$ A painter of mass 70.0 $\mathrm{kg}$ stands on the ladder 3.00 $\mathrm{m}$ from the bottom. Assume the floor is frictionless. Find (a) the tension in the horizontal bar connecting the two halves of the ladder, (b) the normal forces at $A$ and $B$ , and (c) the components of the reaction force at the single hinge $C$ that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also each half of the ladder separately.

Key Concepts

Static Equilibrium

This concept refers to a state in which the sum of all forces and the sum of all moments (torques) acting on a system are zero. It is fundamental in the analysis of structures, as it ensures that an object or system remains at rest or moves with a constant velocity. In problems involving ladders, bridges, or beams, applying the conditions of static equilibrium helps determine unknown forces, including tensions, normal reactions, and other support reactions.

Free-Body Diagram Analysis

A free-body diagram (FBD) is a simplified representation of an object, isolated from its surroundings, with all acting forces and moments shown. The FBD is an essential tool in mechanics as it allows one to visually identify all the forces acting on the system, making it easier to apply the laws of equilibrium. In structural problems, drawing separate free-body diagrams for different parts of the system can be crucial for understanding the internal forces and reactions.

Force Decomposition and Torque Balance

This concept involves breaking down forces into their horizontal and vertical components and ensuring that the net torque about any pivot or point is zero. Torque balance is critical when analyzing extended bodies or systems where moments cause rotation. By choosing appropriate pivot points, one can simplify the calculations, allowing the determination of unknown forces or reactions in complex systems such as multi-component ladders or beams.

Normal and Reaction Forces

Normal forces are support forces exerted perpendicular to the contacting surfaces in a structure, while reaction forces include any additional forces, such as friction or hinge forces, that arise from constraints. These forces are critical in understanding how loads are transferred through structures to their supports. In statics, calculating these forces ensures that structures are properly supported and remain stable under applied loads.

Tension in Structural Members

Tension refers to the pulling force transmitted along a structural member such as a rod, cable, or bar. Understanding the role of tension is essential in analyzing how forces are distributed within a structure, especially when components are connected by elastic or rigid members. Proper analysis of tension helps in ensuring that these components not only support the applied load but also maintain the integrity and stability of the entire structure.

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