Force Constant

Introduction: 

In physics, force constant is an alternative term for a spring constant, as defined by Hooke’s law. This law is a law of physics and is defined as the force required for the extension or compression of a spring by a distance x, varies directly (linearly) concerning that distance, that is, Fs = kx, where k is spring constant and is a characteristic of the spring and x is compression of the spring.

Body : 

What is Force Constant?

As per Hooke’s law, the force required in the compression or enlargement of spring is directionally related or proportional to the distance it is stretched. Force Constant is represented as K. 

The dimension of force constant can be found using the spring force formula i.e

F = – Kx.  

It gives k = – F/x.

The force constant unit (SI) is N.m⁻¹.

In the above force constant formula:

• F is the restoring force of the spring which is directed towards equilibrium
• K is the spring/force constant in (N/m)
• x is the displacement of spring from its equilibrium state
• The negative sign here denotes that the restoring force here is opposite to the displacement

Alternatively, force constant is stated as the force exerted if the displacement is unity in the spring. If force F is considered, that stretches the spring so that it displaces from the equilibrium position by x.

Hooke’s equation (force constant equation) holds in many other situations where an elastic body is deformed, such as a wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body for which the force constant equation is true is said to be linear-elastic or Hookean.

Dimensional Formula of Spring Constant

Spring constant is defined as the restoring force per displacement. The Force is related linearly to the displacement of the system. It can also be defined as the force required for the expansion or compression of a spring is directly proportional to the displacement of the spring.

Mathematically,

We know that,

F = -K/x

K = -F/x

Dimension of Force = MLT⁻²

Dimension of x = L

Therefore the dimension of force constant k is given as

k = -MLT⁻²/L = -MT⁻²

For linear springs

Consider a simple spring helical in nature which has a single end attached to a fixed object and the free end is being pulled by a force whose magnitude is Fs. Suppose that the spring has reached a state of equilibrium, where its length does not change. Let x be the figure by which the free end of the spring is removed from its relaxed position that is when it is not being stretched. Hooke’s law states that

F =  Kx.  

or, equivalently,

k =  F/x.

where k is a real number positive in nature, a characteristic of the spring. The same formula holds when the spring is in a compressed state, with F and x both are negative in that state. According to the force/spring constant formula, the graph of the applied force F as a function of the displacement x will be a straight line that passes through the origin, whose slope is k.

The Spring Constant

The spring constant determines how much force will be needed to deform the spring. The standard international (SI) force constant unit of measurement is the newton/meter, but in North America, they are often measured in pounds/inch. A higher spring constant implies a stiffer spring and similarly, vice versa is also true. The spring constant can be found based on four parameters:

• Wire Diameter: The diameter of the wire containing the spring

• Coil Diameter: The diameter of each coil, measures the tightness of the coil

• Free Length: The length of the spring at rest

• Number of active coils: Number of coils that are free to expand and contract

Relation of Spring Constant and its length

Suppose we have a spring of 6 cm whose spring constant is k. What happens if the spring is splitted into two bits of equal size? There will be a new spring constant for one of these smaller springs, which will be 2k. More generally, the spring constant of a spring is inversely proportional to the length of the spring, assuming we are talking about a specific material spring and thickness.

So let’s say we cut the spring in the above example exactly in two, making two smaller springs each 3 cm in length. A spring constant, twice the original, would apply for smaller springs. This is because it is inversely proportional to the spring constant and the length of the spring. This means that in the smaller spring, the original mass of 30 g would produce only 1 mm of stretch. The larger the spring constant, the smaller the extension that produces a given force.

Conclusion :

1. A higher spring constant implies a stiffer spring, and vice versa is also true.
2. The amount of spring stretch when plotted against the weight and then added to the hanger will give a straight line that runs through the origin. This means that the extension of a spring is directly related or proportional to the pulling force applied to it.
3. In the world of springs, there are many exceptions to Hooke’s law. For example, an extension spring extended too far will not conform to the law. The length at which the spring stops the rule of Hooke’s law is called its elastic limit.