$A$ $1 \, kg$ 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. The frequency of vibration of the $8 \, kg$ block is ..... $Hz$.

Find the time period of mass $M$ when displaced from its equilibrium position and then released for the system shown in the figure.

Two bodies $A$ and $B$ of equal mass are suspended from two massless springs of spring constant $k_1$ and $k_2$,respectively. If the bodies oscillate vertically such that their amplitudes are equal,the ratio of the maximum velocity of $A$ to the maximum velocity of $B$ is

$A$ mass $M$ is suspended by two springs of force constants $K_1$ and $K_2$ respectively,as shown in the diagram. The total elongation (stretch) of the two springs is

An object of mass $m$ is performing simple harmonic motion on a smooth horizontal surface as shown in the figure. Just as the oscillating object reaches its extreme position,another object of mass $2m$ is placed gently on the oscillating object,and it sticks to it. Consider the following statements:
$(a)$ Amplitude of oscillation remains unchanged.
$(b)$ Time period of oscillation remains unchanged.
$(c)$ Total mechanical energy of the system does not change.
$(d)$ The maximum speed of the oscillating object changes.
Which of the above statements are correct?

The time period of a mass suspended from a spring is $T$. If the spring is cut into four equal parts and the same mass is suspended from one of the parts,then the new time period will be