A block of mass $m$ is connected to three springs as shown in the figure. The block is displaced down slightly and left free, it starts oscillating. Find the time period (in sec) of oscillations (neglecting gravity). $\left(m={\pi }^2 \mathrm{kg}, {\pi }^2=10;\theta =60°; k_1=4 \mathrm{N}/\mathrm{m}; k_2=23 \mathrm{N}/\mathrm{m}\right)$

In the given figure, a mass $M$ is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is $k.$ The mass oscillates on a frictionless surface with time period $T$ and amplitude $A.$ When the mass is in equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it. The new amplitude of oscillation will be:

A particle of mass $m$ is attached to four springs with spring constant $k,k,2k$ and $2k$ as shown in the figure. Four springs are attached to the four corners of a square and a particle is placed at the center. If the particle is pushed slightly towards any side of the square and released, the period of oscillation will be

The point $A$ moves with a uniform speed along the circumference of a circle of radius $0.36 \mathrm{m}$ and covers $30°$ in $0.1 \mathrm{s}$. The perpendicular projection $P$ from $A$ on the diameter $MN$ represents the simple harmonic motion of $P$. The restoration force per unit mass when $P$ touches $M$ will be :

In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant $k,$ the frequency of oscillation of given body is:

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

A tray of mass $12 \mathrm{kg}$ is supported by two identical springs as shown in figure. When the tray is pressed down slightly and then released, it executes SHM with a time period of $1.5 \mathrm{s}$. The spring constant of each spring is