16.1 Hooke’s Law: Stress and Strain Revisited
2 min read•june 18, 2024
describes how respond to forces. When stretched or compressed, they push back proportionally to regain their shape. This relationship is key to understanding springs and other elastic objects in physics.
Elastic materials store energy when deformed, following Hooke's Law. This concept connects to broader ideas of force, energy, and material properties. Understanding these principles helps explain everyday objects' behavior under .
Hooke's Law and Elastic Materials
Newton's third law in materials
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- External force applied to a material causes (force per unit area) and (change in shape or size)
- Material exerts an equal and opposite to restore its original shape and size ()
- Elastic materials (springs) return to original shape and size after external force is removed
- (clay) undergo permanent and do not return to original shape and size
Restoration force vs displacement
- In elastic materials, restoration force is directly proportional to from equilibrium position (Hooke's Law: )
- : restoration force ()
- : , measure of material's ()
- : from equilibrium position ()
- Negative sign indicates restoration force acts opposite to displacement direction
- constant is a characteristic property of the elastic material
- Higher means stiffer material, more force required for a given displacement
- Lower means more compliant material, less force required for a given displacement
- This relationship holds true within the material's range
Spring potential energy calculation
- Elastic materials (springs) store potential energy when deformed, known as elastic potential energy
- Elastic potential energy calculated using Hooke's Law and work-energy principle
- Work done to compress or stretch a spring equals change in its elastic potential energy
- Work done is product of average force and displacement:
- Substituting Hooke's Law () into work-energy equation yields elastic potential energy formula:
- : elastic potential energy stored in spring ()
- : spring constant (N/m)
- : displacement from equilibrium position (m)
- Elastic potential energy stored in spring increases quadratically with displacement
- Doubling displacement results in four times the stored elastic potential energy
Material Properties and Limits
- : maximum stress a material can withstand before permanent deformation occurs
- : stress at which a material begins to deform plastically
- : maximum stress a material can withstand before failure
- Different types of elastic moduli describe material behavior under various stresses:
- (tensile or compressive stress)
- (shear stress)
- (uniform pressure from all directions)
Key Terms to Review (37)
AC current: AC current (Alternating Current) is an electric current that reverses its direction periodically. It is the form of electrical energy commonly delivered to businesses and residences.
Approximations: Approximations are simplified representations or calculations that are close to the exact value, used when exact values are impractical. They help in making complex problems more manageable.
Bulk Modulus: Bulk modulus is a measure of a material's resistance to uniform compression. It quantifies how much a material's volume decreases when subjected to a given increase in pressure, and is an important concept in the study of elasticity and Hooke's law.
Compression: Compression is the process of reducing the volume or size of an object or material by applying force. It involves the application of pressure that causes the particles or molecules within a substance to be pushed closer together, resulting in a decrease in the overall size or dimensions of the object.
Deformation: Deformation is the change in shape or size of an object due to applied forces. It can be elastic (reversible) or plastic (permanent).
Deformation: Deformation is the change in the shape or size of an object due to the application of a force. It is a fundamental concept in the study of mechanics, describing how materials respond to external stresses and strains.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It has both magnitude and direction, indicating how far and in what direction the object has moved from its initial position.
Displacement: Displacement is the change in position of an object, measured from a reference point or origin. It describes the straight-line distance and direction an object has moved, without regard to the path taken.
Elastic Limit: The elastic limit is the maximum stress a material can withstand before it begins to deform permanently. It represents the boundary between the elastic and plastic regions of a material's stress-strain curve, marking the point where the material transitions from reversible to irreversible deformation.
Elastic Materials: Elastic materials are substances that can deform under stress but return to their original shape when the stress is removed. This property is crucial for understanding how materials behave under various forces and is closely related to concepts like stress, strain, and Hooke's Law.
Elasticity: Elasticity is the property of a material that enables it to return to its original shape after being deformed by an external force. This behavior is crucial in understanding how materials respond to stress and strain, allowing for practical applications in engineering and materials science. When materials are subjected to forces, elasticity plays a vital role in determining whether they will permanently deform or revert back to their initial state once the force is removed.
F: The symbol 'f' is used to represent various physical quantities in the fields of friction, Hooke's law, and oscillations. It serves as a variable or parameter that helps describe and quantify these important concepts in physics.
Force constant: The force constant, also known as the spring constant, quantifies the stiffness of a spring in Hooke's Law. It is denoted by $k$ and measured in Newtons per meter (N/m).
Hooke's Law: Hooke's law is a fundamental principle in physics that describes the relationship between the force applied to an object and the resulting deformation or displacement of that object. It states that the force required to stretch or compress a spring is proportional to the distance by which the spring is stretched or compressed, within the elastic limit of the material.
Ink jet printer: An inkjet printer is a type of printer that recreates digital images by propelling droplets of ink onto paper. It utilizes electrical charges and fields to control the placement and movement of the ink.
J: J is a fundamental physical quantity that represents the amount of energy transported per unit time, or power. It is a crucial concept in the study of physics, particularly in the context of Hooke's Law, wave energy, and electromagnetic waves.
K: In physics, 'k' represents the spring constant, a key parameter in Hooke's Law that describes the stiffness of a spring. The spring constant indicates how much force is needed to stretch or compress a spring by a unit length, thus connecting the concepts of stress and strain. It also plays a vital role in oscillatory motion, influencing the period and frequency of oscillations, as well as being significant when dealing with electric fields created by multiple charges.
Linear Elasticity: Linear elasticity is a fundamental concept in the study of the mechanical behavior of materials, which describes the linear relationship between the applied stress and the resulting strain within the elastic limit of the material. It forms the basis for understanding the deformation and load-bearing capacity of various structures and components.
M: The variable 'm' is a fundamental quantity that represents mass, a fundamental property of an object that quantifies the amount of matter it contains. This term is central to the understanding of Hooke's Law, oscillations, and the Young's Double Slit Experiment, as mass plays a crucial role in these physical phenomena.
N: N is a variable or constant that represents a specific quantity or value, and it is commonly used in various scientific and mathematical contexts. This term is particularly relevant in the topics of Friction, Variation of Pressure with Depth in a Fluid, Hooke's Law: Stress and Strain Revisited, and Quantum Numbers and Rules, where it serves different purposes and carries distinct meanings.
N/m: N/m, or Newtons per meter, is a unit of measure that represents the force per unit length, commonly used in the context of Hooke's Law to describe the stiffness or spring constant of an object. It is a fundamental unit in the study of stress and strain in materials and structures.
Newton's Third Law: Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. This means that when one object exerts a force on another object, the second object exerts an equal and opposite force on the first. This principle of action and reaction forces is fundamental to understanding the dynamics of various physical systems, from collisions to rocket propulsion.
Plastic Materials: Plastic materials are a class of synthetic or semi-synthetic materials that are highly malleable and can be molded into various shapes and forms. They are composed of long-chain organic polymers that can be manipulated through heat, pressure, or chemical processes to create a wide range of products.
Restoration Force: The restoration force is the force that acts on an object to restore it to its original position or state after it has been displaced or deformed. This concept is particularly relevant in the context of Hooke's Law, which describes the relationship between the applied force and the resulting deformation of an object.
Robert Hooke: Robert Hooke was a 17th-century English scientist known for his foundational work in physics, particularly in the study of elasticity. He formulated Hooke's Law, which states that the strain in a solid is proportional to the stress applied to it, laying the groundwork for understanding materials under deformation. His contributions extend beyond elasticity to fields like biology and mechanics, establishing him as a pivotal figure in early scientific inquiry.
Shear Modulus: The shear modulus, also known as the modulus of rigidity, is a measure of a material's resistance to shear deformation. It quantifies the relationship between the applied shear stress and the resulting shear strain within the elastic range of the material's behavior.
Spring: A spring is an elastic device that stores potential energy when compressed or stretched, returning to its original shape once the force is removed. Springs are often used in various mechanical systems and applications, demonstrating how they can absorb energy and release it as needed. Their behavior can be described through Hooke's Law, which relates the force exerted by the spring to its displacement from the equilibrium position.
Spring Constant: The spring constant, also known as the force constant, is a measure of the stiffness of a spring. It represents the force required to stretch or compress a spring by a unit distance and is a fundamental property of the spring that determines its behavior in various physical contexts.
Stiffness: Stiffness is a measure of the resistance of an elastic body to deformation or displacement under an applied force. It quantifies how difficult it is to bend, stretch, or compress an object, and is a fundamental concept in the study of materials and structures.
Strain: Strain is the measure of deformation representing the displacement between particles in a material body. It is dimensionless and often expressed as a ratio of change in length to original length.
Strain: Strain is a measure of the deformation or change in shape and size of an object or material when a force is applied to it. It quantifies the relative displacement or change in length of an object compared to its original dimensions, and is a dimensionless quantity that describes the amount of stretch or compression experienced by the material.
Stress: Stress is the internal force per unit area within materials that arises from externally applied forces, temperature changes, or other factors. It is usually measured in Pascals (Pa) and calculated as $ \sigma = \frac{F}{A} $ where $ F $ is the force applied and $ A $ is the cross-sectional area.
Stress: Stress refers to the internal force or pressure experienced by a material or object when an external force is applied to it. It is a measure of the intensity of the internal forces acting within a material or structure, which can lead to deformation or failure if the stress exceeds the material's strength.
Tensile Strength: Tensile strength is a measure of a material's ability to withstand pulling or stretching forces without breaking. It is a fundamental property that describes a material's resistance to tension, an important consideration in the design and analysis of structures and materials.
Vector: A vector is a mathematical quantity that has both magnitude (size) and direction. Vectors are used to describe physical quantities in physics, such as displacement, velocity, and force, where both the size and direction of the quantity are important.
Yield Strength: Yield strength is the amount of stress at which a material begins to deform plastically, meaning it will not return to its original shape when the stress is removed. This property is crucial for understanding how materials respond to forces and is directly linked to Hooke's Law, which describes the relationship between stress (force per unit area) and strain (deformation) in elastic materials. Knowing the yield strength helps engineers design structures that can withstand applied loads without permanent deformation.
Young's modulus: Young's modulus is a measure of the stiffness of a material, defined as the ratio of stress (force per unit area) to strain (deformation relative to original length) within the limits of elasticity. It helps in understanding how much a material will deform under a given load and is essential in characterizing how materials respond to forces, linking directly to the concepts of elasticity and Hooke's Law.