In Fig. 12-73, a uniform beam with a weight of 60 N and a length of 3.2 m is hinged at its lower

step 1 In this problem on the topic of equilibrium and elasticity, we are shown a uniform beam which ha

In Fig. 12-73, a uniform beam with a weight of 60 N and a...

In Fig. $12-73,$ a uniform beam with a weight of 60 $\mathrm{N}$ and a length of 3.2 $\mathrm{m}$ is hinged at its lower end, and a horizontal force $\vec{F}$ of magnitude 50 $\mathrm{N}$ acts at its upper end. The beam is held vertical by a cable that makes angle $\theta=25^{\circ}$ with the ground and is attached to the beam at height $h=$ 2.0 $\mathrm{m} .$ What are (a) the tension in the cable and (b) the force on the beam from the hinge in unit-vector notation?

In Fig. $12-73,$ a uniform beam with a weight of 60 $\mathrm{N}$ and a length of
3.2 $\mathrm{m}$ is hinged at its lower end, and a horizontal force $\vec{F}$ of magnitude
50 $\mathrm{N}$ acts at its upper end. The beam is held vertical by a cable that makes
angle $\theta=25^{\circ}$ with the ground and is attached to the beam at height $h=$
2.0 $\mathrm{m} .$ What are (a) the tension in the cable and (b) the force on the
beam from the hinge in unit-vector notation?

Key Concepts

Static Equilibrium

The principle of static equilibrium states that the sum of all forces and the sum of all moments (torques) acting on a body must be zero. This condition is fundamental when analyzing structures and rigid bodies to ensure that they remain at rest without translational or rotational motion.

Moment Equilibrium

Moment equilibrium involves setting the total sum of torques about any chosen pivot point to zero. This concept is crucial in determining unknown forces or moments, as it allows one to balance the turning effects produced by forces acting at various distances from the pivot.

Free-Body Diagram

A free-body diagram is a visual representation that isolates a body and shows all the external forces acting on it. This tool is essential for systematically identifying and applying the equilibrium conditions by mapping out forces such as weights, applied forces, and reaction forces.

Vector Resolution of Forces

In statics, forces are vector quantities with magnitude and direction. Resolving these forces into perpendicular components (typically horizontal and vertical) simplifies the application of equilibrium equations, making it easier to solve for unknown quantities.

Trigonometric Analysis of Forces

Trigonometric methods are used to decompose forces that act at an angle into their horizontal and vertical components. This is particularly important in equilibrium problems where forces like tension in cables must be resolved to accurately balance the net force and moment acting on a structure.

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