Two opposite forces F1=120 N and F2=80 N act on an elastic plank of modulus of elasticity Y=2

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Two opposite forces F1=120 N and F2=80 N act on an elastic...

Two opposite forces $F_{1}=120 \mathrm{~N}$ and $F_{2}=80 \mathrm{~N}$ act on an elastic plank of modulus of elasticity $Y=2 \times 10^{11}$ $\mathrm{N} / \mathrm{m}^{2}$ and length $\ell=1 \mathrm{~m}$ placed over a smooth horizontal surface. The cross-sectional area of the plank is $S=0.5 \mathrm{~m}^{2}$. The change in length of the plank is $x \times$ $10^{-11} \mathrm{~m}$, then find the value of $x$.

Key Concepts

Young's Modulus

Young's modulus is a measure of the stiffness of an elastic material. It relates the stress (force per unit area) applied to a material to the resulting strain (relative deformation). In problems involving stretching or compression of bars, rods, or planks, Young's modulus provides the proportionality constant that allows us to calculate how much the object will elongate or contract under a given force.

Hooke's Law (in the context of elasticity)

Hooke's Law states that, within the elastic limit, the deformation (extension or compression) of an object is directly proportional to the applied force. For a uniform rod or plank, this law is often represented by the formula ?L = FL/(AY), where F is the applied force, L is the original length, A is the cross-sectional area, and Y is the Young's modulus. This relationship is fundamental in determining how elastic materials respond to forces.

Stress and Strain

Stress is defined as the force applied per unit area, and strain is the measure of deformation expressed as the change in length over the original length. The relationship between stress and strain in elastic materials is linear, as described by Young's modulus. This concept provides the basis for understanding how materials deform under load and is essential for solving problems involving elastic deformation.

Internal Force Distribution in a Free Body

When a free body, such as an elastic plank on a frictionless surface, is subjected to unequal forces at its ends, the object experiences a net acceleration. However, the internal forces that cause deformation are determined by the forces acting at the boundaries that resist the change in length. In such cases, the effective force that stretches the object is the one that is transmitted through the material (often the smaller magnitude force), and this principle is used alongside Hooke's law to determine the resultant deformation.

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