Figure P 8.61 shows a lightweight plank supported at its...

Figure $\mathrm{P} 8.61$ shows a lightweight plank supported at its right end by a 7.0 -mm-diameter rope with a tensile strength of $6.0 \times 10^{7} \mathrm{N} / \mathrm{m}^{2}.$ a. What is the maximum force that the rope can support? b. What is the greatest distance, measured from the pivot, that the center of gravity of an $800 \mathrm{kg}$ piece of heavy machinery can be placed without snapping the rope?

Figure $\mathrm{P} 8.61$ shows a lightweight plank supported at its right end by a 7.0 -mm-diameter rope with a tensile strength of $6.0 \times 10^{7} \mathrm{N} / \mathrm{m}^{2}.$
a. What is the maximum force that the rope can support?
b. What is the greatest distance, measured from the pivot, that the center of gravity of an $800 \mathrm{kg}$ piece of heavy machinery can be placed without snapping the rope?

Key Concepts

Center of Mass

The center of mass is the point where the mass of an object or system is considered to be concentrated, and it is at this point that gravitational force effectively acts. In problems involving rotational equilibrium, understanding the position of the center of mass relative to the pivot is crucial for accurately calculating the moments or torques that result from the weight of the object.

Torque Equilibrium

Torque equilibrium is a fundamental principle in statics that states that for a system to be in balance, the sum of the clockwise moments about any pivot point must equal the sum of the counterclockwise moments. This principle is essential when calculating how far a load’s center of mass can be positioned from a pivot without causing rotation or failure in the supporting structure.

Tensile Strength

Tensile strength is a material property that indicates the maximum stress that a material can withstand while being stretched or pulled before failing. In contexts involving ropes, cables, or other fibers, it is used to determine how much load the material can support before breaking, based on the product of the tensile strength and the cross?sectional area of the material.

Cross?Sectional Area Calculation

The calculation of the cross?sectional area is crucial when determining the load-bearing capability of a material such as a rope. For circular cross sections, the area is determined using the formula A = ?r² (or A = ?d²/4 when using diameter), which, when multiplied by tensile strength, gives the maximum force that the material can safely support.

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