A helical spring of stiffness k is cut into two halves and a mass m is connected to the two halves.

The natural time period of this system is found to be 0.5 s. If an identical spring is cut so that one part
is one-fourth and the other part three-fourths of the original length, and the mass m is connected to the
two parts. Find the natural period of the system?

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To find the natural period of the system, we need to consider the equation of the simple harmonic motion of a mass-spring system:

T = 2π√(m/k)

where T is the period, m is the mass, and k is the stiffness of the spring.

In the first scenario, the helical spring is cut into two halves, and the mass m is connected to the two halves. Let's call the stiffness of each half of the spring k₁.

In this case, the effective stiffness k′ will be the sum of the two halves:

k' = k₁ + k₁ = 2k₁

Since the natural time period of the system is given as 0.5 s, we can write the equation as:

0.5 = 2π√(m/k')

Now, let's consider the second scenario where an identical spring is cut such that one part is one-fourth and the other part three-fourths of the original length. Let the stiffness of each half be k₂.

In this case, the effective stiffness k'' will also be the sum of the two halves:

k'' = k₂ + k₂ = 2k₂

To find the natural period of the system in this scenario, we can write the equation as:

T'' = 2π√(m/k'')

However, we need to determine the relationship between k' and k'' in order to find T''.

Given that one part is one-fourth and the other part is three-fourths of the original length, the stiffness of each half is inversely proportional to their respective lengths. Let L be the original length of the spring.

Therefore, k''/k' = (3/4L)/(1/4L) = 3/1 = 3

Substituting this relationship into the equation for T'', we have:

T'' = 2π√(m/(3k'))

Since we know that T = 0.5 s for the first scenario, we can substitute this value into the equation to solve for k':

0.5 = 2π√(m/k')
√(m/k') = 0.25/π
m/k' = (0.25/π)^2
k' = m/π^2

Now, substituting this expression for k' into the equation for T'':

T'' = 2π√(m/(3k'))
= 2π√(m/(3(m/π^2)))
= 2π√((π^2m)/(3m))
= 2π√(π^2/3)
= 2π(π/√3)
= 2π^2/√3

Therefore, the natural period of the system in the second scenario is 2π^2/√3.