Question:
A spring when connected by mass m gives time period T if spring is cut into n equal parts and each part connected in parallel with same mass m gives time period will be
Answer:
Using the Ideal spring Equation F = -kx, where F is the force applied, k is the spring constant and x is the compressed distance.
The time period of a harmonic oscillator when mass m and spring constant kare constant is T=2π (m/k)1/2
Let the ratio of spring constants of the three spring segments = 1/3:1/3:1/3
So consider the values to be x/3. When these three are in series, they give rise to the original spring. So
(x/3 * x/3 * x/3) / (x/3 + x/3 + x/3) = k
which gives x2/27 = k.which means x = (27k)1/2
So spring constants are (27k)1/2
When they are in parallel, effective spring constant = (27k)1/2 + (27k)1/2 + (27k)1/2
= 3 * (27k)1/2
Time period T =
2 π m / [3 * (27k) power ½ ]
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