Question

The spring constants of three springs connected to a mass M are shown in figure. When mass oscillates, what are the effective spring constant and the time period of vibration?

• A. $4k,2\pi \sqrt{\frac{M}{4k}}$
• B. $3k,2\pi \sqrt{\frac{M}{3}}k$
• C. $2k,2\pi \sqrt{\frac{M}{2k}}$
• D. None of these

Answer 1

Answer: (D)

None of these

In lower segment/zone, two springs, each of constant k, are in parallel.
$\therefore$ Equivalent spring constant $=(k+k)=2k$
Now, for the system, there is a spring (of constant $2k$) in upper segment and a spring (of constant $2k$) in lower segment.
They are in series
$\therefore k_r=\frac{k_1{k}_2}{(k_1+k_2)}=\frac{\left(2k\right)\left(2k\right)}{(2k+2k)}$
$\therefore k_r=4k^2/4k=k$ and
$\therefore T=2\pi \sqrt{\frac{M}{k}}$.