$A$ mass $m$ attached to the free end of a spring executes $SHM$ with a period of $1\; s$. If the mass is increased by $3\; kg$,the period of oscillation increases by $1\; s$. The value of mass $m$ is $..............kg$.

$A$ block $A$ of mass $1\, kg$ is connected to two identical springs of spring constant $800\, N/m$ and is placed on a smooth horizontal surface as shown in the figure. Initially,the springs are relaxed. Now,the block $A$ is slightly displaced to the left and released. The time period of oscillation of the system is

On a frictionless horizontal plane,a bob of mass $m=0.1 \text{ kg}$ is attached to a spring with natural length $l_0=0.1 \text{ m}$. The spring constant is $k_1=0.009 \text{ N/m}$ when the length of the spring $l > l_0$ and is $k_2=0.016 \text{ N/m}$ when $l < l_0$. Initially,the bob is released from $l=0.15 \text{ m}$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=(n \pi) \text{ s}$,then the integer closest to $n$ is:

Two identical springs of spring constant $2k$ are attached to a block of mass $m$ and to fixed supports (see figure). When the mass is displaced from the equilibrium position on either side,it executes simple harmonic motion. The time period of oscillations of this system is ...... .

Two small bodies of mass $2 \, kg$ each are attached to each other using a thread of length $10 \, cm$,and hang on a spring whose force constant is $200 \, N/m$,as shown in the figure. We burn the thread. What is the distance between the two bodies when the top body first arrives at its highest position? (Take $\pi^2 = 10$)

The potential energy of a particle of mass $0.1\,kg,$ moving along the $x-$axis,is given by $U = 5x(x-4)\,J,$ where $x$ is in metres. It can be concluded that