A uniform, horizontal, 100-N pipe is used as a lever, as...

A uniform, horizontal, $100-\mathrm{N}$ pipe is used as a lever, as shown in Fig. $5-4 .$ Where must the fulcrum (the support point) be placed if a 500-N weight at one end is to balance a 200-N weight at the other end? What is the upward reaction force exerted by the support on the pipe? The forces in question are shown in Fig. $5-4$, where $F_{R}$ is the reaction force of the support on the pipe. The weight of the pipe acts downward at its center. We assume that the support point is at a distance $x$ from one end. Take the axis of rotation to be at the support point. Then the torque equation, $f_{+} \sum \tau=0$, about that point becomes $$ +(x)(200 \mathrm{~N})\left(\sin 90^{\circ}\right)+(x-L / 2)(100 \mathrm{~N})\left(\sin 90^{\circ}\right)-(L-x)(500 \mathrm{~N})\left(\sin 90^{\circ}\right)=0 $$ This simplifies to $$ (800 \mathrm{~N})(x)=(550 \mathrm{~N})(L) $$ and so $x=0.69 L$. The support should be placed $0.69$ of the way from the lighter-loaded end. To find $F_{R}$ use $+\uparrow \sum F_{y}=0$, $$ F_{R}-200 \mathrm{~N}-100 \mathrm{~N}-500 \mathrm{~N}=0 $$ from which $F_{0}=800 \mathrm{~N}$

A uniform, horizontal, $100-\mathrm{N}$ pipe is used as a lever, as shown in Fig. $5-4 .$ Where must the fulcrum (the support point) be placed if a 500-N weight at one end is to balance a 200-N weight at the other end? What is the upward reaction force exerted by the support on the pipe?

The forces in question are shown in Fig. $5-4$, where $F_{R}$ is the reaction force of the support on the pipe. The weight of the pipe acts downward at its center. We assume that the support point is at a distance $x$ from one end. Take the axis of rotation to be at the support point. Then the torque equation, $f_{+} \sum \tau=0$, about that point becomes
$$
+(x)(200 \mathrm{~N})\left(\sin 90^{\circ}\right)+(x-L / 2)(100 \mathrm{~N})\left(\sin 90^{\circ}\right)-(L-x)(500 \mathrm{~N})\left(\sin 90^{\circ}\right)=0
$$
This simplifies to
$$
(800 \mathrm{~N})(x)=(550 \mathrm{~N})(L)
$$
and so $x=0.69 L$. The support should be placed $0.69$ of the way from the lighter-loaded end.
To find $F_{R}$ use $+\uparrow \sum F_{y}=0$,
$$
F_{R}-200 \mathrm{~N}-100 \mathrm{~N}-500 \mathrm{~N}=0
$$
from which $F_{0}=800 \mathrm{~N}$

Key Concepts

Center of Mass

The center of mass is a point that represents the average location of the mass distribution in an object. In uniform or symmetric bodies, it often lies at the geometric center. Understanding the center of mass is important because it allows one to simplify the problem by treating the distributed weight as if it were concentrated at a single point, aiding in the analysis of equilibrium conditions.

Reaction Force

The reaction force is the force exerted by a support or surface to counterbalance the applied loads, ensuring overall equilibrium. It is determined by applying the condition of translational equilibrium, where the sum of the upward reaction forces must equal the sum of the downward forces, thereby preventing vertical acceleration of the system.

Translational Equilibrium

Translational equilibrium is achieved when the sum of all forces acting on a system is zero, meaning that the system experiences no net linear acceleration. This condition ensures that, aside from rotational considerations, the entire body remains at rest or moves with a constant velocity, which is fundamental in static analysis.

Torque (Moment of Force)

Torque is the measure of the turning effect of a force, calculated as the product of the force magnitude and the perpendicular distance from the pivot or axis of rotation. It is essential in analyzing rotational equilibrium because for a system to remain stable, the sum of the clockwise torques must exactly balance the sum of the counterclockwise torques.

Lever Principle

This concept involves using a rigid bar that rotates about a fixed point (the fulcrum) to balance forces applied at different distances from the fulcrum. The lever principle underlies how forces can be amplified or balanced by adjusting their application points relative to the pivot, which is central to understanding how mechanical advantage is achieved in static systems.

Rotational Equilibrium

Rotational equilibrium occurs when the sum of all torques acting on a system about any axis equals zero, ensuring that the system does not undergo angular acceleration. This principle is crucial when determining unknown distances or forces in lever problems, as it provides the condition required for balance about the fulcrum.

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