Two different, massless springs are connected to each other and then vertically

Question

Two different, massless springs are connected to each other and then vertically suspended from the ceiling. A particle, of mass M, is then attached to the bottom of the lowest spring. The particle is subsequently displaced from its equilibrium position and set into simple harmonic motion. Derive and equation for the peridot T of the springs motion. Two different, massless springs are connected to each other and then vertically suspended from the ceiling. A particle, of mass M, is then attached to the bottom of the lowest spring. The particle is subsequently displaced from its equilibrium position and set into simple harmonic motion. Derive and equation for the peridot T of the springs motion.

Explanation / Answer

let displacement of spring one be x1

and displacement of spring two be x2

so force = Keq* (x1 + x2) ........... (a)

now tension in both springs must be same so that they dont buckle

so K1x1 = K2x2

x1 => K2*x2/ K1 .......(b)

so equation (a) becomes

f = Keq((K2+K1)/K1) * x2

also f = K2x2

so ,

K2x2 = Keq((K2+K1)/K1) * x2

Keq = K1*K2 /(K1 + K2)

T = 2* pi * sqrt(m/ Keq)

=2 Pi sqrt((K1 + K2) / (K1 * K2))

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