Two identical springs (spring constant = k ) are connected to two light rigid rods in parallel combination and then a third spring (identical to the other two) is connected to this combination in series. The combined system is fixed to the ceiling and a block of mass m is attached as shown in the figure. If the mass is slightly displaced and then let go, then the system shall oscillate with a time period of

Question

Two identical springs (spring constant = k ) are connected to two light rigid rods in parallel combination and then a third spring (identical to the other two) is connected to this combination in series. The combined system is fixed to the ceiling and a block of mass m is attached as shown in the figure. If the mass is slightly displaced and then let go, then the system shall oscillate with a time period of <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-sucm1nnp2jtawde1.jpg" /> (A) 2π √(m/3 k) (B) 2π √(3 m/2 k) (C) 2π √(2 m/3 k) (D) 2π √(3 k/m)

Tardigrade Admin · Accepted Answer

Consider the system as shown in figure below.

Here,

k_{2}

and

k_{3}

are in parallel.

k_{1}

is in series with parallel combination of

k_{2}

and

k_{3} .

Effective spring constant of the combination is

k_{e ff} = \frac{( k _{2} + k _{3} ) k _{1}}{k _{1} + k _{2} + k _{3}}

= \frac{2 k \times k}{3 k} = \frac{2}{3} k

∴

Time period of oscillation

T = 2 π \frac{m}{k _{e ff}}

= 2 π \frac{m}{\frac{2}{3} k}

= 2 π \frac{3 m}{2 k}

$2 π \frac{m}{3 k}$

$2 π \frac{3 m}{2 k}$

$2 π \frac{2 m}{3 k}$

$2 π \frac{3 k}{m}$

2π3km​​

2π2k3m​​

2π3k2m​​

2πm3k​​

$2 π \frac{m}{3 k}$

$2 π \frac{3 m}{2 k}$

$2 π \frac{2 m}{3 k}$

$2 π \frac{3 k}{m}$