Dimension of Force Constant

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What is Spring Constant?

According to Hooke's law, the force required to compress or enlarge a spring is directionally proportional to the distance it is stretched. The spring Constant is represented as K. 

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

F = -Kx.  

It gives k = -F/x.

The SI unit of spring constant is N.m⁻¹.


Here in the above spring constant formula:

F is the restoring force of the spring directed towards the equilibrium.

K is the spring constant in (N/M)

X is represented as the displacement of the spring from its equilibrium position.


The negative symbol states that the restoring force is opposite to the displacement.

It is represented in Newton per meter (N/m)


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


Dimensional Formula of Spring Constant

Spring constant is stated as the restoring force per displacement. Force is linearly related to the displacement of the system. It can also be stated that the force required to extend or compress 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]

Hence, the dimension of force constant k is given as

k = -[MLT⁻²/][L] = -[MT⁻²]


State the Theorem “ The Dimension of Force Constant is MT⁻²”.

Proof

From Hooke’s Law, we have

F = -K/x

Here,

F is a force of direction, displacement MLT⁻²

x is a displacement, of direction L

Let the dimension of K be D

Then we have,

MLT⁻² = DL

The result obtained from the above equation is

D = MT⁻²

Hence the theorem, the dimension of force constant is proved.


Can a Spring Constant Consider a Value Zero or it Can be Negative?

Based on Hooke's Law,

F = -K/x

The negative sign in the above indicates that the force applied by the string will always be on the opposite side of the force applied by the load. It is because the string always tries to take the position of its original length. The spring should always have a positive value as it represents the stiffness of spring. 

If the spring constant is zero, it implies that the stiffness of spring is zero. It will no longer be considered as spring as there will be no force acting in the opposite direction. If the spring constant were considered to be a negative value, it implies that instead of an equal and opposite force, the spring will always act in the direction of displacement. It implies that even if spring is attached by a single unit, it will continue to stretch infinitely.


Solved Examples

1. Calculate the Spring Constant if the Spring with Load 5 cm is Stretched by 40 cm.

Solution:

Given,

Mass (m )= 5 kg

Displacement (x )= 40 cm

We know that

Force (F) = m.a.

= 5 × p.4 = 2N

The spring constant is stated as

K= -F/X

= -2/0.4

= -5 N/m


2. A Girl Weighing 20 Pounds Stretched a Spring by 50 cm. Calculate the Spring Constant of the Spring.

Solution:

Given,

Mass (m) = 20 lbs 

= 20 /2.2

= 9.09 kg

Displacement (x) = 50 cm

The force = ma = 9.09 * 9.08

= 89.082 N

The spring constant formula is derived as

k = -F/x

= 89.082/ 0.5

= - 178.164 N/m

FAQ (Frequently Asked Questions)

1. What is Hooke's Law?

Ans: Hooke’s law states that the force needed to enlarge or compress a spring by some distance is directly proportional to that distance. The rigidity of spring is a constant factor characteristic. The property of the elastic states that it takes twice the force to stretch a spring twice longer. The linear dependence of displacement of stretching is known as Hooke’s Law.


This law is named after the 17th Century British Physicist Robert Hooke. It states that the amount of stress we apply on any object is equivalent to the amount of strain observed on it which implies stress-strain.

2. What is a Constant Force Spring?

Ans: A constant force spring is a spring for which the force it applies over the range of motion is constant. The constant force spring does not follow Hooke’s law. Generally, constant force springs are formed as a rolled ribbon of spring steel such that the string is rested when it is completely rolled up. As the spring is unrolled, the restoring force occurs primarily from the portion of the ribbon near the roll. Since the shape of that portion remains constant as the spring unrolls, the resultant force is approximately constant. 


A conical spring that is formed is said to have a constant rate by forming the spring with a variable pitch. Placing a bigger spring in a bigger OD coil and smaller spring in a smaller OD coil will exert the spring to disintegrate all the coils at similar rates when compressed. 


Some of the applications of constant force spring can be seen in door closers, cables, retractors, counterbalances, cabinet furniture components, hairdryers, toys, electric motors, appliances, space vehicles, and gym equipment.