Question

A mass mis suspended from a spring of force constant k and just touches another identical spring fixed to the floor as shown in figure. The time period of small oscillations is:

Answer 1

T = 2π√(m/2k)

Answer 2

When the mass is at equilibrium, the top spring is stretched by $mg/k$ and its free end just touches the bottom spring. For small oscillations, if the mass is displaced by a distance $x$, then:

• The top spring is stretched an extra $x$ (restoring force $kx$ upward),
• The bottom spring is compressed by $x$ (restoring force $kx$ upward).

Thus, the net restoring force is:

$$F = -kx -kx = -2kx,$$

which means the effective spring constant is $k_{\text{eff}} = 2k$.

The time period for small oscillations is then given by:

$$T = 2\pi\sqrt{\frac{m}{k_{\text{eff}}}} = 2\pi\sqrt{\frac{m}{2k}}.$$