Question

what is Spring factor derive the expression for resultant Spring Constant when two springs having constants K1 and K2 are connected in parallel and series​

Answer 1

When two massless springs following Hooke's Law, are connected via a thin, vertical rod as shown in the figure below, these are said to be connected in parallel. Spring 1 and 2 have spring constants

k

1

and

k

2

respectively. A constant force

→

F

is

This system of two parallel springs is equivalent to a single Hookean spring, of spring constant

k

. The value of

k

can be found from the formula that applies to capacitors connected in parallel in an electrical circuit.

k

=

k

1

+

k

2

Series.

When same springs are connected as shown in the figure below, these are said to be connected in series. A constant force

→

F

is applied on spring 2. So that the springs are extended and the total extension of the combination is the sum of constant

k

. The value of

k

can be found from the formula that applies to capacitors connected in series in an electrical circuit.

For spring 1, from Hooke's Law

F

=

k

1

x

1

where

x

1

is the deformation of spring.

Similarly if

x

2

is the

Answer 2

$\huge{\star}{\underline{\boxed{\red{\sf{Answer :}}}}}{\star}$

$\Large{\underline{\underline{\sf{Series :}}}}$

When a body is pulled down a less distance y , and the two springs have different extensions y1 and y2 .

$\Large{\sf{F \: = \: -k_{1}y_{1}}--(1)}$

And

And $\Large{\sf{F \: = \: -k_{2}y_{2}}--(2)}$

$\rule{50}{2}$

(1) and (2) can be written as,

$\Large{\boxed{\sf{y_{1} \: = \: \frac{-F}{k_{1}}}}}$

And

$\Large{\boxed{\sf{y_{2} \: = \: \frac{-F}{k_{2}}}}}$

As,

y = y1 + y2

$\Large \leadsto {\sf{y \: = \: \frac{-F}{k_{1} \: - \: \frac{F}{k_{2}}}}}$

Take L.C.M

$\Large \leadsto {\sf{y \: = \: -F(\frac{k_{1} \: + \: k_{2}}{k_{1}k_{2}})}}$

$\LARGE \implies {\sf{F \: = \: - (\frac{k_{1}k_{2}}{k_{1} \: + \: k_{2}})y}}$

As K is springs constant

$\Large {\boxed{\sf{K \: = \: \frac{k_{1}k_{2}}{k_{1} \: + \: k_{2}}}}}$

Now, Put Values in Time period of S.H.M

$\LARGE{\boxed{\boxed{\sf{T \: = \: 2 \pi \sqrt{\frac{Inertia \: Factor}{Spring \: Factor }}}}}}$

$\Large \leadsto {\sf{T \: = \: 2 \pi \sqrt{\frac{m}{K}}}}$

Put Value of K

$\huge{\boxed{\boxed{\sf{T \: = \: 2 \pi \sqrt{\frac{m(k_{1} \: + \: k_{2})}{k_{1}k_{2}}}}}}}$