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# Combination of springs in SHM , Practice problem, FAQs

Take a spring and fix the one end of spring to the wall and pull the other end upto some distance. Now take another spring, fix the one end of this spring to the second end of the previous spring; On pulling springs to the same distance you will feel that the force you are appling is lesser than the force in the first case. This arrangement of spring is called a series combination of spring. Moving forward, if you fix the one end of the second spring to the wall and pull the ends of both springs then you will feel greater force than the first case.This arrangement of springs is called a parallel combination of springs.Let understand why it is happening?

Table of content

• Cutting of a spring
• Combination of springs
• Series combination
• Parallel combination
• Practice problem
• FAQs

## Cutting of a spring

Consider a massless spring having natural length L and spring constant K as shown in the figure. If A force F is applied on the spring, and the extension in the string is x then,

F=kx...(i)

Now if we assume this spring as a elastic string of cross section area A, then stress and strain in the string

YA= constant so we can say

$k\propto \frac{1}{L}kL=constant$

kL=constant

From this we can conclude that the product of spring constant and its length is constant. If a spring is cut into ‘n’ identical pieces, then the spring constant of each piece is nk.

Example - If a spring of length l0 and spring constant k is cut into three equal parts, then the spring constant of each part becomes 3k.

## Combination of springs

There are mainly two combinations of springs, namely series and parallel.

When one end of one spring is connected to another end of the second spring, it becomes a series combination. In a parallel combination, one end of the spring is connected to a rigid support, and the other end is connected to a block; the second spring also has the same configuration.

## Series combination

The series combination of springs is shown in figures below. In a series combination of springs, the extension in the springs is different for the force of the same magnitude. The extension in the springs depends on the stiffness.

For the same force F, the extensions will be x1 and x2 having spring constants k1 and k2.

Then

$F={k}_{1}{x}_{1}andF={k}_{2}{x}_{2}.......\left(i\right)$

Suppose that the whole system is replaced by a single spring of spring constant keq and the net extension x occurs as shown in the figure.

Then

$F={k}_{eq}{x}_{.}......\left(ii\right)$

As total extension x will be equal to the sum of extension of individual spring i.e.

$x={x}_{1}+{x}_{2}$

Substitute the values of x,x1 and x2 from equation (i) and (ii), we get

$\frac{F}{{k}_{eq}}=\frac{F}{{k}_{1}}+\frac{F}{{k}_{2}}\phantom{\rule{0ex}{0ex}}\frac{1}{{k}_{eq}}=\frac{1}{{k}_{1}}+\frac{1}{{k}_{2}}$

This is the formula for the force constant of an equivalent spring of springs in series.

Instead of having two, if we have n number of spring in series, then equivalent spring constant is given as,

$\frac{1}{{k}_{eq}}=\frac{1}{{k}_{1}}+\frac{1}{{k}_{2}}+\frac{1}{{k}_{3}}..............+\frac{1}{{k}_{n}}$

The time period of oscillation of the combination is given by,

$T=2\pi \sqrt{\frac{m}{{k}_{eq}}}$

## Parallel combination

The parallel combinations of springs are shown in the figure.If the parallel combination of two massless springs is stretched up to a distance by some external force, then the extension of both the springs will be the same.

Suppose a force F is applied to the block then it will distribute on both the springs. let the individual springs of spring constants k1 and k2 feel the force F1 and F2, respectively.

Then

F=F1+F2

Where F1=k1x and F2=k2x

Hence, F=k1x+k2x........(i)

Now, suppose that the whole system is replaced by a single spring of spring constant keq and the same amount of extension occurs due to the same force F as shown in the figure.

Then

This is the formula for the force constant of an equivalent spring of springs in parallel.

Instead of having two springs, if we have n number of spring in Parallel, then equivalent spring constant is given as,

${k}_{eq}={k}_{1}+{k}_{2}+{k}_{3}...........+{k}_{n}$

The time period of oscillation of the combination is given by,

$T=2\pi \sqrt{\frac{m}{{k}_{eq}}}$

## Practice problem

Q. The force constant of the two springs in series are k1= 1 N/m and k2= 2 N/m. Find the equivalent spring constant.

A. Given k1= 1 N/m and k2= 2 N/m

Equivalent spring constant is given as

Q.Two similar springs of force constant k are connected in series. If a block of mass m is suspended from them, then find the frequency of oscillation.

A. As springs are connected in series so equivalent spring constant

$\frac{1}{{k}_{eq}}=\frac{1}{{k}_{1}}+\frac{1}{{k}_{2}}\phantom{\rule{0ex}{0ex}}\frac{1}{{k}_{eq}}=\frac{1}{k}+\frac{1}{k}\phantom{\rule{0ex}{0ex}}\frac{1}{{k}_{eq}}=\frac{2}{k}\phantom{\rule{0ex}{0ex}}{k}_{eq}=\frac{k}{2}$

Frequency of oscillation is given as

$f=\frac{1}{2\pi }\sqrt{\frac{{k}_{eq}}{m}}\phantom{\rule{0ex}{0ex}}f=\frac{1}{2\pi }\sqrt{\frac{k}{2m}}Ans$

Q. Two springs are connected in parallel if the force constant of springs are k1= 9 N/m and k2= 6 N/m, then what is the equivalent spring constant.

A. Given k1= 9 N/m and k2= 6 N/m

In case of parallel combination equivalent spring constant is

Q. A metal plate having mass M = 10 kg is supported by two parallel springs as shown in the figure. If the plate is given a small displacement it starts a simple harmonic motion with a time period of 2 s. Now a block of mass m is put on a plate, Its time period becomes 6 s. Find the mass of the block.

A.

The time period of the spring block system is given as,

## FAQs

Q. Why is the force applied on two springs in series less than as applied to the individual spring?
A.
In series combination springs, the equivalent force constant is less, so the applied force F=keqx is also less.

Q. If a spring is cut into two equal parts, the spring constant of the individual part is?
A.
We can treat the original spring as a combination of two parts in series. So, the spring constant of individual parts will be double the original spring constant.

Q. Can the value of force constant be negative?
A.
No, force constant is always positive.

Q. What do you understand by spring stiffness?
A.
It is a constant which relates the resistance to extension or compression of springs with load.

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