Hooke’s Law Formula

Hooke’s Law Formula

Hooke’s Law is a fundamental principle in physics that describes the behavior of elastic materials, specifically how they deform under applied forces. Named after the 17th-century English scientist Robert Hooke, the law states that the force needed to extend or compress a spring by some distance is directly proportional to that distance. In mathematical terms, िफ़ F represents the force applied to the spring and x denotes the displacement (change in length) from its equilibrium position, Hooke’s Law can be expressed as F=kx, where k is the spring constant—a measure of the stiffness of the spring. Learn more about Hooke’s Law in this article by Extramarks

What is Hooke’s Law?

According to Hookes Law Formula, for relatively minor deformations of an object, the displacement or magnitude of the deformation is directly proportional to the deforming force or load.

Hooke’s law governs the opposing force created by a spring. Robert Hooke, an English scientist, created Hooke’s law of elasticity. It does not matter what kind of material is deformed, or how much. Hooke’s law only applies to elastic materials in the elastic area (up to the proportionate limit). This means that Hooke’s law will not hold true if any elastic material is stretched or compressed above the proportionate limit.

Hooke’s Law Formula

Hooke’s Law formula is given as

F = -kx

where

• F is restoring force
• k is  spring constant
• x is change in length

Derivation of Hooke’s law

To derive Hooke’s Law, let’s consider a spring that obeys this fundamental principle of elasticity. Hooke’s Law states that the force F required to extend or compress a spring by a displacement x from its equilibrium position is directly proportional to that displacement.

The spring is considered ideal and obeys Hooke’s Law within its elastic limit, meaning it returns to its original shape and size when the applied force is removed.

When a spring is stretched or compressed, it exerts a restoring force that opposes the displacement. This force is directly proportional to how far the spring is stretched or compressed from its equilibrium position.

Consider a spring with a spring constant k. If the spring is stretched or compressed by a distance x from its equilibrium position, the restoring force F is proportional to x.

The spring constant k depends on the material of the spring, its geometry, and other physical properties. To derive Hooke’s Law, we assume that for a small displacement x, the force F is given by F=kx.

Applications of Hooke’s Law

Hooke’s Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position, finds numerous practical applications across various fields.

• Spring Design and Manufacturing:
  • Hooke’s Law is fundamental in designing and manufacturing springs used in mechanical systems. It helps engineers determine the required spring stiffness (spring constant $k$) to achieve specific displacements and forces within a system.
• Mechanical Systems and Devices:
  • Springs based on Hooke’s Law are extensively used in various mechanical systems and devices to provide suspension, shock absorption, and oscillation damping. Examples include car suspensions, watches, mattresses, and bicycle suspensions.
• Force Measurement:
  • Hooke’s Law is utilized in force measurement devices such as spring scales. The deflection of the spring due to the applied force directly indicates the magnitude of the force according to F=kx.
• Materials Testing:
  • It is used in materials testing laboratories to determine the mechanical properties of materials, including Young’s modulus, which relates stress to strain in materials subjected to tensile or compressive forces.

Limitations of Hooke’s Law

While Hooke’s Law provides a useful approximation for many elastic materials and systems, especially within their linear range of deformation, it has several limitations that are important to consider:

• Limited Applicability to Non-linear Materials:
  • Hooke’s Law assumes that the relationship between stress (force per unit area) and strain (deformation) is linear. However, many materials exhibit non-linear behavior at higher stresses or strains, where the relationship between force and displacement deviates from a straight line.
• Elastic Limit:
  • Hooke’s Law is applicable only up to the elastic limit of a material. Beyond this limit, the material may undergo plastic deformation or even fracture, where the relationship between stress and strain is no longer linear.
• Temperature Dependency:
  • The stiffness (spring constant $k$) of a material, as described by Hooke’s Law, can vary with temperature. At higher temperatures, materials may become less stiff or even exhibit viscoelastic behavior, where the rate of deformation affects the stress-strain relationship.

Hooke’s Law Solved Examples

Example 1: A spring with a spring constant \( k = 200 \) N/m is stretched by 0.1 meters from its equilibrium position. Calculate the restoring force exerted by the spring.

Solution:

According to Hooke’s Law, the restoring force \( F \) exerted by a spring is given by:
\[ F = kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

Given:
Spring constant, \( k = 200 \) N/m
Displacement, \( x = 0.1 \) m

Substitute the values into the formula:
\[ F = 200 \times 0.1 \]
\[ F = 20 \text{ N} \]

The restoring force exerted by the spring is \( 20 \) N.

Example 2: A force of \( 50 \) N stretches a spring by \( 0.2 \) meters. Determine the spring constant \( k \).

Solution:

Using Hooke’s Law \( F = kx \), where \( F \) is the force applied, \( x \) is the displacement, and \( k \) is the spring constant:

Given:
Force, \( F = 50 \) N
Displacement, \( x = 0.2 \) m

To find \( k \):
\[ k = \frac{F}{x} \]
\[ k = \frac{50}{0.2} \]
\[ k = 250 \text{ N/m} \]

The spring constant \( k \) is \( 250 \) N/m.

Example 3: A spring with a spring constant \( k = 100 \) N/m is compressed by \( 0.15 \) meters. Calculate the restoring force exerted by the spring.

Solution:

Using Hooke’s Law:
\[ F = kx \]

Given:
Spring constant, \( k = 100 \) N/m
Displacement, \( x = 0.15 \) m (negative because it’s compression)

Substitute the values:
\[ F = 100 \times (-0.15) \]
\[ F = -15 \text{ N} \]

The negative sign indicates that the force is acting in the opposite direction of the compression (towards equilibrium).

The restoring force exerted by the spring is 15 N (in the direction opposite to compression).