The correct option is
D T=2π√m2kSuppose the particle is displaced by a distance
x downward. Then, the upper two springs will be elongated and lower springs will be compressed. The force will act as shown in the FBD given below:
Let
y be the elongation in the upper springs. The net restoring force by the upper springs is
F′=F1cos(90−θ)+F2cos(90−θ)
By symmetry, we can say that,
F1=F2=F (say)
⇒F′=2Fsinθ .....(1)
Let the lower springs be compressed by
y′ each.
The net restoring force by the two lower springs is given by
F′′=F3cos(90−β)+F4cos(90−β)
By symmetry, we can say that
F3=F4=f (say)
⇒F′′=2fsinβ ......(2)
From
(1) and
(2), we get
Fnet=2Fsinθ+2fsinβ
Fnet=−(2kysinθ+4ky′sinβ)
From the first diagram, we can deduce that
Elongation in upper spring
y=xcos(90−θ)=xsinθ
Elongation in lower spring
y′=xcos(90−β)=xsinβ
Therefore,
Fnet=−(2kxsin2θ+4kxsin2β)
From the data given in the question,
Fnet=−[2kx12+4kx14]
Fnet=−2kx⇒a=−2kxm
Comparing this with
a=−ω2x, we get
ω=√2km⇒T=2π√m2k
Thus, option (d) is the correct answer.