Figure 12-81 shows a 300 kg cylinder that is horizontal....

Figure $12-81$ shows a 300 $\mathrm{kg}$ cylinder that is horizontal. Three steel wires support the cylinder from a ceiling. Wires 1 and 3 are attached at the ends of the cylinder, and wire 2 is attached at the center. The wires each have a cross-sectional area of $2.00 \times 10^{-6} \mathrm{m}^{2}$. Initially (before the cylinder was put in place) wires 1 and 3 were 2.0000 $\mathrm{m}$ long and wire 2 was 6.00 $\mathrm{mm}$ longer than that. Now (with the cylinder in place) all three wires have been stretched. What is the tension in (a) wire 1 and (b) wire 2?

Figure $12-81$ shows a 300 $\mathrm{kg}$ cylinder that is horizontal. Three
steel wires support the cylinder from a ceiling. Wires 1 and 3 are attached at the ends of the cylinder, and wire 2 is attached at the center. The wires each have a cross-sectional area of $2.00 \times 10^{-6} \mathrm{m}^{2}$. Initially (before the cylinder was put in place) wires 1 and 3 were 2.0000 $\mathrm{m}$ long and wire 2 was 6.00 $\mathrm{mm}$ longer than that. Now (with the cylinder in place) all three wires have been stretched. What is the tension in (a) wire 1 and (b) wire 2?

Key Concepts

Mechanical Equilibrium

This concept involves applying the conditions that the sum of all forces and the sum of all moments (torques) in a system must equal zero when the system is in a state of rest. In problems involving suspended bodies, analyzing equilibrium is essential to determining how forces like gravity, tension, and support reactions balance one another.

Tension and Force Distribution

In systems with multiple supporting elements such as wires or cables, the overall weight of the object is distributed among the supports. Understanding how to calculate the individual tensions in each support requires analyzing how forces are shared based on the geometry of the setup and the points of attachment.

Hooke’s Law and Elasticity

Hooke’s Law relates the force exerted on an elastic material to its extension or compression through a proportionality constant. This law is central to determining how much a wire or spring stretches under a load, which in turn affects the tension in the material.

Stress, Strain, and Young’s Modulus

When a material is subjected to a force, it experiences stress (force per unit area) and strain (the relative deformation). Young’s modulus is the constant of proportionality linking stress and strain in the elastic (linear) region of the material's response. This concept is critical for calculating how much a wire will stretch under a given load and thus helps in determining the tension in the wire.

Transcript

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