Where must a 0.80-kN object be hung on a uniform, horizontal, rigid 100-N pole so that a girl pu

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Where must a 0.80-kN object be hung on a uniform,...

Where must a $0.80-\mathrm{kN}$ object be hung on a uniform, horizontal, rigid $100-\mathrm{N}$ pole so that a girl pushing up at one end supports one-third as much as a woman pushing up at the other end? The situation is shown in Fig. 5-5, where the weight of the pole acts down at its center. We represent the force exerted by the girl as $F$, and that by the woman as $3 F$. There are two unknowns, $F$ and $x$, and we will need two equations. To avoid the possibility of writing equations that turn out not to be independent, it's a good practice to write one sum-of-the-torques equation and one sum-of-the-forces equation. Take the rotational axis point at the left end. Then the torque equation becomes $$ -(x)(800 \mathrm{~N})\left(\sin 90^{\circ}\right)-(L / 2)(100 \mathrm{~N})\left(\sin 90^{\circ}\right)+(L)(F)\left(\sin 90^{\circ}\right)=0 $$ For the second equation write $$ +1 \sum F_y=3 F-800 \mathrm{~N}-100 \mathrm{~N}+F=0 $$ from which $F=225 \mathrm{~N}$. Substitution of this value in the torque equation yields $$ (800 \mathrm{~N})(x)=(225 \mathrm{~N})(L)-(100 \mathrm{~N})(L / 2) $$ and so $x=0.22 L$. The load should be hung 0.22 of the way from the woman to the girl.

Where must a $0.80-\mathrm{kN}$ object be hung on a uniform, horizontal, rigid $100-\mathrm{N}$ pole so that a girl pushing up at one end supports one-third as much as a woman pushing up at the other end?

The situation is shown in Fig. 5-5, where the weight of the pole acts down at its center. We represent the force exerted by the girl as $F$, and that by the woman as $3 F$. There are two unknowns, $F$ and $x$, and we will need two equations. To avoid the possibility of writing equations that turn out not to be independent, it's a good practice to write one sum-of-the-torques equation and one sum-of-the-forces equation. Take the rotational axis point at the left end. Then the torque equation becomes

$$
-(x)(800 \mathrm{~N})\left(\sin 90^{\circ}\right)-(L / 2)(100 \mathrm{~N})\left(\sin 90^{\circ}\right)+(L)(F)\left(\sin 90^{\circ}\right)=0
$$

For the second equation write

$$
+1 \sum F_y=3 F-800 \mathrm{~N}-100 \mathrm{~N}+F=0
$$

from which $F=225 \mathrm{~N}$. Substitution of this value in the torque equation yields

$$
(800 \mathrm{~N})(x)=(225 \mathrm{~N})(L)-(100 \mathrm{~N})(L / 2)
$$

and so $x=0.22 L$. The load should be hung 0.22 of the way from the woman to the girl.

Key Concepts

Simultaneous Equations in Statics

In many statics problems, the conditions of force equilibrium and torque equilibrium provide a system of simultaneous equations. Solving these equations allows for the determination of unknown forces and distances essential for maintaining the object in static equilibrium. This methodical approach enables precise analysis of complex mechanical systems.

Center of Mass in Uniform Objects

For uniform objects, the weight (or center of mass) is evenly distributed, meaning that the gravitational force can be considered to act at the midpoint of the object. This simplifies the analysis of torques since the weight can be assumed to be concentrated at that specific point, providing a consistent method for applying gravitational forces in equilibrium calculations.

Lever Arm

A lever arm is the perpendicular distance from the axis of rotation (pivot) to the line of action of a force. Accurate identification and calculation of lever arms is important for computing the torque produced by forces. This concept underpins the lever principle, which is central to analyzing moments and balances in statics.

Force Equilibrium

Force equilibrium means that the total vertical (or horizontal) forces in a system cancel each other out, resulting in no net acceleration. In problems involving supports or loads, setting the sum of the upward forces equal to the sum of the downward forces is critical to finding unknown forces and ensuring that the structure remains stationary.

Static Equilibrium

This concept involves ensuring that a system is at rest with both the net force and net torque equal to zero. In a statics problem, such as balancing forces and torques on a beam or pole, the conditions for static equilibrium allow one to set up mathematical equations that govern the forces acting on the object and the moments produced about any chosen pivot point.

Torque (Moment) Equilibrium

Torque equilibrium refers to the condition that the sum of all torques (moments) about any point in the system must be zero. This is essential in determining unknown distances or forces, as each force creates a turning effect on the object. By choosing an appropriate pivot point, the problem can often be simplified by eliminating the torque contribution of particular forces.

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