Introduction To Spring Constant

In the last article, we learnt the basic concepts of SHM. Now we shall try to visualise the spring-mass system. Before that, we will see what spring constant is. Force by the action of the spring is given by,

\(F\) = \(-kx\)

k is known as the spring constant or stiffness constant.

Unit of spring constant is N/m.

There are different types of spring. For example torsion spring which works due to turning of the spring.

We can also visualise this spring-mass motion with the help of uniform circular motion.

Spring Constant

Spring-mass motion using uniform circular motion

Suppose at \( t\) =\( 0 \) the particle is as shown in the figure. If we take the projection of that on the x-axis it is at origin. At \(t \) =\( T_1\), if we take the projection along x-axis it is in positive x-direction and having positive velocity. Similarly, at \(t\) = \(T_2\) , the projection is in negative x-axis and having negative velocity. By observing this we can say we take the projection of uniform circular motion on the x-axis it represents an SHM. Sometimes this approach is very useful in solving the numerical problem as compared to equation-based approach.

How to find spring constant when a number of springs are connected together? Suppose we have a situation as shown below:

Spring ConstantSpring-mass system 1

Suppose the mass is displaced by x towards right, so the force exerted by S2

\(F_2\) = \(F_2=-k_2x\) (Towards left)

Force exerted by \(S_1\) ,

\(F_1\) = \(-k_1x\) (Towards left)

So the total restoring force \((F)\) =\( F_1 + F_2\)

\(F\) = \(- (k_1 + k_2) x\)

Hence, equivalent \(k \) = \(k_1 + k_2\)

Using a similar approach try to find equivalent spring constant for the following?

Spring Constant

Spring-mass system 2

Also, the time period for the spring mass system can be shown to be,

\(T\) = \(2π\sqrt{\frac{m}{k}}\)

Where k is the equivalent stiffness constant.

Stay tuned with BYJU’S to learn more about spring constant, SHM and much more.

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