A 15,000-N crane pivots around a friction-free axle at...

$\mathrm{A} \quad 15,000-\mathrm{N} \quad$ crane pivots around a friction-free axle at its base and is supported by a cable making a $25^{\circ}$ angle with the crane (Fig. Ell.18). The crane is 16 $\mathrm{m}$ long and isnot uniform, its center of gravity being 7.0 $\mathrm{m}$ from the axle as measured along the crane. Thecable is attached 3.0 $\mathrm{m}$ from the upper end of the crane. When the crane is raised to $55^{\circ}$ above the horizontal holding an $11,000-\mathrm{N}$ pallet of bricks by a $2.2-\mathrm{m}$ , very light cord, find (a) the tension in the cable and (b) the horizontal and vertical components of the force that the axle exerts on the crane. Start with a free-body diagram of the crane.

$\mathrm{A} \quad 15,000-\mathrm{N} \quad$ crane pivots around a friction-free axle at its base and is supported by a cable making a $25^{\circ}$ angle with the crane (Fig. Ell.18). The crane is 16 $\mathrm{m}$ long and isnot uniform, its center of gravity being 7.0 $\mathrm{m}$ from the axle as
measured along the crane. Thecable is attached 3.0 $\mathrm{m}$ from the upper end of the crane.
When the crane is raised to $55^{\circ}$ above the horizontal holding an $11,000-\mathrm{N}$ pallet of bricks by a $2.2-\mathrm{m}$ , very light cord, find (a) the tension in the cable and (b) the horizontal and vertical components of the force that the axle exerts on the crane. Start with a free-body diagram of the crane.

Key Concepts

Static Equilibrium

Static equilibrium refers to the state of a system where the sum of all forces and the sum of all torques (moments) acting on the system are zero. This concept is crucial for analyzing any structure or object at rest, ensuring that the object will not translate or rotate. In problems involving cranes or beams, applying the equilibrium conditions (?F = 0 and ?? = 0) helps in solving for unknown forces such as reaction forces and tensions.

Free-Body Diagram

A free-body diagram (FBD) is a schematic representation that isolates an object and displays all the external forces and moments acting on it. Constructing an FBD is an essential step in solving static equilibrium problems because it allows for a clear identification of the forces, such as weight, tension, and reaction forces, as well as their points of application and directions.

Torque and Moment Arms

Torque, also known as a moment, is the product of a force and its moment arm (the perpendicular distance from the pivot point to the line of action of the force). In static equilibrium problems, setting up a torque balance about a convenient pivot point enables the calculation of unknown forces by eliminating the contributions of forces that pass through the pivot. Understanding how to calculate and use moment arms is crucial in determining forces in structures like cranes.

Center of Gravity

The center of gravity (CG) is the point at which the entire weight of a body can be considered to act. For non-uniform objects, the CG might not be at the geometric center, and its location is critical for computing torques accurately. The position of the CG affects the stability and balance of structures, making it an important factor in equilibrium analysis.

Force Components and Decomposition

Forces that act at an angle can be decomposed into horizontal and vertical components using trigonometric functions. This decomposition is essential when applying the equilibrium conditions separately in the horizontal and vertical directions. Understanding how to resolve forces into components helps in analyzing the reaction forces and the net effect of combined forces on the equilibrium of the system.

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