A $1 \,kg$ block attached to a spring vibrates with a frequency of $1\, Hz$ on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an $8\, kg$ block placed on the same table. So, the frequency of vibration of the $8\, kg$ block is ….. $Hz$

A block of mass $2\,kg$ is attached with two identical springs of spring constant $20\,N / m$ each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $\frac{\pi}{\sqrt{x}}$ in SI unit. The value of $x$ is $……….$

Infinite springs with force constant $k$, $2k$, $4k$ and $8k$…. respectively are connected in series. The effective force constant of the spring will be

A body of mass $m$ is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand, so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is $4\,cm$ below the point, where it was held in hand.

$(a)$ What is the amplitude of oscillation ?

$(b)$ Find the frequency of oscillation.

The springs in figure. $A$ and $B$ are identical but length in $A$ is three times that in $B$. The ratio of period $T_A/T_B$ is