Three masses $700g, 500g$, and $400g$ are suspended at the end of a spring a shown and are in equilibrium. When the $700g$ mass is removed, the system oscillates with a period of $3$ seconds, when the $500 \,gm$ mass is also removed, it will oscillate with a period of ...... $s$

The springs shown are identical. When $A = 4kg$, the elongation of spring is $1\, cm$. If $B = 6\,kg$, the elongation produced by it is  ..... $ cm$

When a mass $m$ is attached to a spring it oscillates with period $4 \,s$. When an additional mass of $2 \,kg$ is attached to a spring, time period increases by $1 \,s$. The value of $m$ is ........... $kg$

If the period of oscillation of mass $m$ suspended from a spring is $2\, sec$, then the period of mass $4m$ will be  .... $\sec$

For the damped oscillator shown in Figure the mass mof the block is $200\; g , k=90 \;N m ^{-1}$ and the damping constant $b$ is $40 \;g s ^{-1} .$ Calculate

$(a)$ the period of oscillation,

$(b)$ time taken for its amplitude of vibrations to drop to half of Its inittal value, and

$(c)$ the time taken for its mechanical energy to drop to half its initial value.

If a spring has time period $T$, and is cut into $n$ equal parts, then the time period of each part will be