 # Spring Constant

## Spring Constant Units

Spring constant definition is related to simple harmonic motions and Hooke's law. So, before we try to define spring constant and understand the workings of spring constant, we need to look at Hooke's law. According to the theory of elasticity, when a load is applied o a spring it will naturally extend proportionally, as long as the load applied is less than the elastic limit. Now we know that when force is applied on to an object, it tends to deform in some way. Consider a spring, when we apply force on one side of the spring, it will get compressed, as they are elastic. At this time the spring exerts its force in the direction opposite to the applied force, to expand to its original size. Therefore, to define spring constant, we first define Hooke's law. Hooke's law is defined as the force required by the spring to revert to its size is directly proportional to the distance of the compression of the spring.

The image shows the movement of the spring when force is applied to one side.

### Spring Constant Definition

To understand the spring constant definition, we will look at the Hooke’s law formula. Hooke’s law formula is also known as the spring constant formula. The formula is given below.

 Formula F = -kx Spring Constant Units Nm-1

Where F represents the restoring force of the spring, x is the displacement of the spring, and k is known as the spring constant. The spring constant units are given as Newton per meter.

Now, that we know that k is the spring constant, we will look at spring constant definition. We define spring constant as the stiffness of the spring. In other words, when the displacement of the spring is one unit, we can define spring constant as the force applied to cause that said displacement. Therefore, it is clear to say that, the stiffer the spring is, the higher will be its spring constant.

### Spring Constant Dimensional Formula

According to Hooke’s law, we know that,

F= -kx

Therefore,

k= -F/x

Now we know that the unit of force is given as Newton (N), or as kg m/s2.

Therefore, we can write the dimensional unit as [MLT-2].

We also know the dimensional unit of x is given as [L]

Applying spring constant formula, we get,

k = - $\frac{[MLT^{-2}]}{[L]}$

k= - [MT-2]

The spring constant unit is in terms of Newton per meter (N/m).

### Solved Problems

Question 1) A spring is stretched by 40cm when a load of 5kg is added to it. Find the spring constant.

Mass m = 5kg,

Displacement x = 40cm = 0.4m

To find the spring constant, we first need to find the force that is acting on the spring.

We know that F = m * x

Therefore, F = 5 * 0.4

F = 2N

The load applies a force of 2N on the spring. Hence the spring will apply an equal and opposite force of – 2N.

Now, by substitute the values in the spring constant formula we get,

k = -F/x

k = - $\frac{-2}{0.4}$

k = 5 N/m

Therefore, the spring constant of the spring is 5N/m

Question 2) Consider a spring with a spring constant of 14000N/m. A force of 3500N is applied to the spring. What will be the displacement of the spring?

Force F = 3500N,

Spring constant k = 14000N/m,

We can calculate the displacement of the spring by using the spring constant formula.

x= -F/k

The load applies a force of 3500N on the spring. Hence the spring will apply an equal and opposite force of – 3500N.

Thus,

x = -(- 3500/14000)

x = 0.25 m

x = 25 cm

Therefore, the spring is displaced by a distance of 25cm.

Can Spring Constant Take A Value Of Zero, or Can It Be Negative?

According to Hooke’s law,

F= -kx

Here the negative sign tells us that the force applied by the spring will always be on the opposite side of the force applied by the load. This is because the spring will always try to get to its original length. The spring constant represents the stiffness of the spring; hence it should always have a positive value.

If the spring constant is zero, it means that the stiffness of the spring will be zero. It will no longer be a spring as no force will be acting in the opposite direction. Now if the spring constant were to be a negative value, it would mean that instead of an equal and opposite force, the spring will apply a force in the direction of the displacement. Which means that, even if the spring is stretched by a single unit, it will continue stretching infinitely.

2. What Happens To The Value Of Spring Constant When Several Springs Are Connected In Series Or Parallel?

When springs are connected in series the force acting on both the springs will be equal, which is in turn equal to the external force.

Consider two springs connected in series with spring constant k1 and k2 respectfully. A force F is applied to one side of the spring. We know that,

k = -F/x

X1 is the displacement of spring 1, and X2 is the displacement of spring 2.

X = X1 + X2

Therefore,

(F/k) = (F/k1) + (F/k2)

(1/k) = (1/k1) + (1/k2)

When springs are connected in parallel, the force acting on both the springs will be divided, but the deflection will be the same.

Consider two springs connected in series with spring constant k1 and k2 respectfully. A force F is applied to one side of the spring. We know that,

F = -kx

F1 is the force acting on spring 1, and F2 is the force acting on spring 2.

F = F1 + F2

Therefore,

kx = k1x + k2x

k = k1 + k2.