The differential equation of a simple harmonic motion is given by $\frac{d^2y}{dt^2} + ky = 0$. What is the time period?

$A$ particle moves such that its acceleration $a$ is given by $a = -bx$,where $x$ is the displacement from the equilibrium position and $b$ is a constant. The period of oscillation is

The displacement of a particle varies with time according to the relation $x = a \sin \omega t + b \cos \omega t$.

The motion of a particle is described by the equation $a = -bx$,where $a$ is the acceleration,$x$ is the displacement from the equilibrium position,and $b$ is a constant. The periodic time will be

What provides the restoring force in the following cases?
$(1)$ Compressed spring.
$(2)$ Displacement of water in a $U$-tube.
$(3)$ Displacement of a pendulum bob from its mean position.

The equation of $SHM$ of a particle is given as $2 \frac{d^2x}{dt^2} + 32x = 0$,where $x$ is the displacement from the mean position of rest. The period of its oscillation (in seconds) is