Spring Constant Formula: Hooke’s Law, Potential Energy, & Solved Examples

Spring constant’s meaning is associated with simple harmonic motions and Hooke's law. As per Hooke’s law, the force needed to extend or compress a spring is directly proportional to the distance it is stretched.

Simple harmonic motion (SHM) is used in the oscillatory motion of objects. Springs have their own natural “spring constants” which specify how inflexible they are. Hooke's law defines SHM and provides a formula for the applied force using the spring constant. 

Also check: Work Energy and Power

Key Terms: Spring Constant, Hooke’s Law, Spring Constant Formula, Potential Energy of a Spring

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What is Spring Constant (or Hooke’s Law)

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Hook’s law is specifying the connection between the distance stretched and the force applied in the spring. It states that the force needed to extend or compress a spring is directly proportionate to the distance it is stretched.

The spring constant defines how much force will be required to deform the spring. The standard international (SI) force constant unit of measurement is Newton/meter, but it is often measured in pounds/inch in North America. A higher spring constant indicates a stiffer spring and similarly, the reverse is also true. The spring constant can be discovered based on four parameters:

• Wire Diameter: The diameter of the wire that contains the spring.
• Coil Diameter: Each coil’s diameter estimates the tightness of the coil.
• Free length: The spring’s length at rest.
• A number of Active Coils: The number of coils that are free for expansion and contraction.

The letter K is the spring constant and has units as N/m. According to Newton's third law of motion, when a spring is stretched, it retracts with a restoring force. This restoring force follows Hooke's law, which relates the spring force to the spring constant.

Force of the spring (F) = - (spring constant (k)) * (displacement (x))

• F = – k × x
• Spring Constant Units = N.m^-1

Where

• F = the restoring force of the spring directed toward equilibrium.
• k = the spring constant in N.m^-1.
• x = displacing the spring from its equilibrium point.

In other words, the spring constant is the applied force if the displacement in the spring is unity. If a force F is assumed to stretch the spring so that it displaces the equilibrium position by x.

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Relation of Spring Constant and its length

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Assume we have a 6 cm spring with spring constant k. Now, if the spring is split into two bits of equal size, a new spring constant for one of these smaller springs will be 2k. More commonly, the spring constant of a spring is inversely proportionate to the length of the spring, considering we are talking about a special material spring and thickness.

So, let's say we cut the spring in the above example precisely in half, creating two smaller springs of length 3 cm each. The spring constant, twice the original, will apply to smaller springs. The reason behind this is that it is inversely proportional to the spring constant and the length of the spring. This indicates that in a small spring, an original mass of 30 grams will produce only 1 mm of stretch. The bigger the spring constant, the smaller the extension that produces the mentioned force.

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Spring Constant Dimensional Formula

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We know that,

F= – kx

Therefore,

\(k = –\frac{F}{x}\)

Dimension of F = [MLT^-2]

Dimension of x = [L]

Therefore, the dimension of k =

\(k=-\frac{[MLT^{-2}]}{[L]}=-[MT^{-2}]\)

The Spring Constant Formula is given as,

\(k = –\frac{F}{x}\)

Where,

• F = Force applied,
• x = displacement by the spring
• A negative sign indicates that the restoring force is opposite to the displacement.

It is described in Newton per meter (N/m).

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Potential Energy of a Spring (P.E.)

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The energy stored in a stretchable or compressible object is known as the spring’s potential energy. It is also known as elastic potential energy and is equivalent to the force multiplied by the distance traveled.

It is referred as: Potential Energy = Force × Displacement

Also force of the spring is equal to the spring constant × displacement. So,

P.E. = \(\frac{1}{2}\) KX² ⇢ (2)

The above-mentioned equation is the formula for the potential energy of a spring.

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Applications of Hooke’s Law

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• Because of the elasticity of springs, Hooke's law is usually applied to springs.
• It is used not only in the field of engineering but also in the field of medical science.
• It is useful in diving boards, spring beds, lungs, skin, and automobile suspension systems.
• It is the basic principle underlying spring scales, manometers, and clock balance wheels.
• It is also the basis of acoustics, seismology, and molecular mechanics.

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Limitations of Hooke’s Law

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The drawbacks of Hooke's law are as follows:

• Hooke's law only applies in the elastic region then it fails.
• Hooke's law gives correct results only for solid bodies with small forces and deformations.

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Things to Remember

• As per Hooke’s law, the force needed to extend or compress a spring is directly proportional to the distance it is stretched.
• Force of the spring (F) = - (spring constant (k)) * (displacement (x))
  • F = – k × x
  • Spring Constant Units = N.m^-1

Dimension of k =\(k=-\frac{[MLT^{-2}]}{[L]}=-[MT^{-2}]\)

• The energy stored in a stretchable or compressible object is known as spring potential energy. P.E. = \(\frac{1}{2}\) KX² ⇢ (2)
• Hooke's law has the limitation that it applies only under the elastic limit of any material, which means that the material must be perfectly elastic to obey Hooke's law.
• Because of the elasticity of springs, Hooke's law is usually applied to springs. It is used not only in the field of engineering but also in the field of medical science.
• Hooke's law only applies in the elastic region then it fails and gives correct results only for solid bodies with small forces and deformations.

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