A 20.0 -m-long uniform beam weighing 550 N rests on walls A and B, as shown in Fig. 9-80 .(a) Fi

step 1 Question number 68, the free body diagram of the beam is like this. At point A, the reaction is

A 20.0 -m-long uniform beam weighing 550 N rests on walls A...

A 20.0 -m-long uniform beam weighing 550 $\mathrm{N}$ rests on walls $\mathrm{A}$ and $\mathrm{B},$ as shown in Fig. $9-80 .(a)$ Find the maximum weight of a person who can walk to the extreme end $\mathrm{D}$ without tipping the beam. Find the forces that the walls $A$ and $B$ exert on the beam when the person is standing: $(b)$ at $D ;(c)$ at a point 2.0 $\mathrm{m}$ to the right of $\mathrm{B} ;(d) 2.0 \mathrm{m}$ to the right of $\mathrm{A}$ .

A 20.0 -m-long uniform beam weighing 550 $\mathrm{N}$ rests on walls $\mathrm{A}$ and $\mathrm{B},$ as shown in Fig. $9-80 .(a)$ Find the maximum weight of a person who can walk to the extreme end $\mathrm{D}$ without tipping the beam. Find the forces that the walls $A$
and $B$ exert on the beam when the person is standing: $(b)$ at $D ;(c)$ at a point 2.0 $\mathrm{m}$ to the right of $\mathrm{B} ;(d) 2.0 \mathrm{m}$ to the right of $\mathrm{A}$ .

Key Concepts

Lever Arm

A lever arm is the perpendicular distance from the axis of rotation to the line of action of a force. It plays a key role in determining the magnitude of the torque generated by a force. In the context of beam problems, understanding lever arms allows for accurate calculations of moments created by different forces, such as the weight of the beam or external loads like a person standing on it.

Reaction Forces at Supports

Reaction forces are the forces exerted by supports or connections that keep a structure in equilibrium. In beam problems, these forces are determined by applying the conditions of static equilibrium and ensuring that the sum of moments and vertical forces are zero. They are crucial for assessing the load-bearing capacity of the supports and ensuring that the beam remains stable under various applied loads.

Static Equilibrium

Static equilibrium is a fundamental concept in mechanics, ensuring that an object at rest has all forces and torques balanced. For a beam or any rigid body, this means that the sum of all vertical forces and the sum of all moments (or torques) about any pivot point must equal zero. This principle allows for the solving of unknown forces acting on the body when it is subjected to various loads.

Torque (Moment) Equilibrium

Torque equilibrium involves balancing the turning effects of forces around a pivot point. In problems involving beams or levers, calculating the moments (product of force and the distance from the pivot) is essential in ensuring that the body does not rotate. Setting the clockwise moments equal to the counterclockwise moments helps determine unknown forces or positions where loads can be applied without tipping the system.

Center of Mass (Uniform Beam)

For a uniform beam, the weight distribution is even, and its entire weight can be thought of as acting at its center of mass, typically located at the midpoint of the beam. This simplification is critical in calculating the moment due to the beam's own weight, as it provides a single point where the gravitational force is considered to be applied.

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