What does the slope of an applied-force-versus-displacement graph represent for a linear spring?

The spring constant $k$, which measures stiffness.

What does a stiffness matrix represent in linear elasticity?

A linear mapping from a displacement vector to a force vector, including directional coupling.

What distinguishes elastic recovery from linear proportionality?

A material can recover its shape while its stress–strain response is not linear.

How does the axial stiffness of a uniform elastic rod depend on its dimensions?

It is $k=AE/L$, where $A$ is cross-sectional area, $E$ is Young's modulus, and $L$ is length.

Mechanical engineeringLesson 4 of 5 in the course Strength of materials

Hooke's Law: Springs, Elasticity, and Material Response

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Hooke's law describes how an elastic object responds to a small force. It is a foundational model for understanding springs and many engineering materials.

For a spring displaced a distance $x$ from its relaxed position, the magnitude of the force needed to hold it there is

Here, $F_s$ is the applied force, $x$ is the extension or compression, and $k$ is the spring constant, or stiffness. A larger $k$ means more force is required for the same displacement. Rearranging gives $x=F_s/k$ .

All flashcards

What does the slope of an applied-force-versus-displacement graph represent for a linear spring?: The spring constant $k$ , which measures stiffness.
How is elastic potential energy related to the displacement of an ideal linear spring?: It is $U_{el}=rac{1}{2}kx^2$ .
What does a stiffness matrix represent in linear elasticity?: A linear mapping from a displacement vector to a force vector, including directional coupling.
What distinguishes elastic recovery from linear proportionality?: A material can recover its shape while its stress–strain response is not linear.
How does the axial stiffness of a uniform elastic rod depend on its dimensions?: It is $k=AE/L$ , where $A$ is cross-sectional area, $E$ is Young's modulus, and $L$ is length.

Quiz questions

For a torsional spring with angular displacement $\theta$ , which equation gives its restoring torque?Answer: $\tau=-k\theta$
Which pair gives the two independent elastic moduli used to describe an isotropic linear-elastic material?Answer: Bulk modulus $K$ and shear modulus $G$
How many independent elastic coefficients describe a general anisotropic material in three-dimensional linear elasticity?Answer: 21
A uniform rod has length $L$ , area $A$ , and Young's modulus $E$ . What is its equivalent axial spring stiffness?Answer: $k=AE/L$
For an ideal mass–spring oscillator with mass $m$ and stiffness $k$ , which expression gives its frequency?Answer: $f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Based on the Wikipedia article Hooke's law, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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