Hooke’s Law Equations

Hooke’s law is a law of physics that is used to solve problems in the field of elasticity. Hooke’s law equations prove that the force (F) required to expand or contract a spring by some distance has linear measure with respect to the distance.

The basic Hooke’s law equation is Fs = kx. Where k is a constant element of the spring and x is small in comparison to the distortion of the spring. An elastic material for which Hooke’s law equations can be assumed is called linear – elastic or Hookean. This law is a precise approximation for most solid bodies. The condition is that the forces and deformations should be small. 

Derivation of Hooke’s law equations

This is the derivation of the equation of Hooke’s law for linear springs. Consider a normal helical spring. One end of the spring is attached to a fixed object and another end is pulled by a force. The magnitude of this force is Fs. Assume that the spring has reached the state of equilibrium, where its length is not varying. Let x be the quantity by which the free end of the spring was substituted from its relaxed position. The relaxed position is the position where the spring is not stretched. 

Hooke’s law equation states that Fs = kx or x = Fs / k, where k is a positive real number. This formula is the same for both expansion and contraction of spring. This leads Fs and x both negative in that case. According to Hooke’s law equation, the graph of the force (Fs) as a function of displacement (x) will be a straight line. This straight line will pass through the origin. The slope of this line will be k. 

Fs is the restoring force that is exerted by the spring on the thing that is pulling its free end. In this case, Hooke’s law equation becomes, Fs = -kx. This is because the direction of the restoring force is opposite to the displacement.  

Hooke’s law equations in the concepts of stress and strain

Stress

The term stress is determined as the force per unit area. It is the ratio of applied force to a cross-sectional area. There are three types of stress. 

Tensile stress

This type of stress leads to a stretch of the material. It acts standard to the stressed region. 

Formula: σ = Fn / A where σ is the normal stress, Fn is the force and A is the area. 

Compressive stress

This type of stress compresses the material. It acts standard to the stressed region. The formula is the same as tensile stress. 

Shearing stress. 

This type of stress cuts the material. It acts in the plane to the stressed area. Shearing stress acts perpendicular to compressive or tensile stress. 

Formula : T = Fp / A

Strain

Strain is termed as deformation or distortion of a solid because of stress. There are two types of strain.

Normal strain

This type of strain elongates or contracts the line segment.

Formula: e = dl/l0, where e is the strain, dl is the change in length and l0 is the initial length. 

Shear strain

This type of strain changes the angle between two line segments with right angle.

Hooke’s law equations importance with real-life examples

Masses and springs are common in physics. These serve as a typical method to explain and apply Hooke’s law. Here are some real-life examples of Hooke’s law equations.

• Heavy cargo pushing a vehicle to settle, when the compression system. This leads to the lowering of the vehicle towards the ground. 
• A door banging into a doorstep. 
• A withdrawable pen that uses a spring. 
• The recoil of a toy gun which uses spring

Applications of Hooke’s law equations

• Hooke’s law equations are applied in strings due to their use in elasticity. 
• These equations are used in engineering, medical sciences, etc. 
• It is used as a basic concept in the manometer, balance wheel on a clock, and a spring scale
• These equations are the base of acoustics, molecular mechanics, and seismology.

Conclusion

Hooke’s law equations state that the force required to expand or contract a spring by some distance estimates linearly with respect to the distance. Hooke’s law equations prove that the force (F) required to expand or contract a spring by some distance has linear measure with respect to the distance. The formula for Hooke’s law equation is Fs = kx. Where k is a constant element of the spring and x is small in comparison to the distortion of the spring. Hooke’s law equations are applied in strings due to their use in elasticity. These equations are used in engineering, medical sciences, etc.