A uniform beam is 5.0 m long and has a mass of 53 kg . In Fig. 12- 74, the beam is supported in

step 1 So here we're going to say that counterclockwise is positive, and we can then say that the sum o

A uniform beam is 5.0 m long and has a mass of 53 kg . In...

A uniform beam is 5.0 $\mathrm{m}$ long and has a mass of 53 $\mathrm{kg} .$ In Fig. $12-$ $74,$ the beam is supported in a horizontal position by a hinge and a cable, with angle $\theta=60^{\circ} .$ In unit-vector notation, what is the force on the beam from the hinge?

Key Concepts

Center of Mass

The center of mass of an object is the point at which its mass can be considered to be concentrated for the purpose of analysis. In the context of uniform objects, like a beam with evenly distributed mass, the gravitational force can be assumed to act at the center of mass, which is typically at its geometric center. This is crucial for calculating the torque due to weight.

Force Decomposition in Unit Vectors

Force decomposition involves resolving a force into its components along specified directions, typically using unit vectors (such as i, j, and k in the Cartesian coordinate system). This technique simplifies the analysis by allowing one to treat each directional component independently when applying equilibrium conditions.

Static Equilibrium

Static equilibrium is a fundamental concept in mechanics where an object remains at rest because the sum of all forces and the sum of all torques (moments) acting upon it are zero. This principle is used to analyze structures and bodies, ensuring that all applied forces, including gravitational, tension, and reaction forces, balance out so that there is no net translational or rotational motion.

Torque (Moment) Equilibrium

Torque equilibrium is the condition that the sum of all torques about any chosen pivot point is zero for a system in static equilibrium. It involves calculating the turning effect of forces acting at various distances from the pivot. This concept is crucial when determining unknown forces or reaction components by setting the net moment to zero, thereby ensuring the system remains rotationally stable.

Free Body Diagram

A free body diagram is a simplified illustration of an object isolated from its surroundings, where all external forces, moments, and reaction forces acting on it are depicted. This tool is essential for visualizing and setting up the equations of equilibrium, as it helps in systematically identifying and breaking down all forces and moments acting on the body.

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