As illustrated in the figure, a rectangular plate with sides a and...

When the rectangular plate is given a small displacement within its plane such that the strings remain taut, the two strings, the ceiling, and the top edge of the plate form a parallelogram.

Because the opposite sides of this linkage are equal (the strings both have length $L$, and the distance between them is $d$ at both the ceiling and the plate), the plate undergoes pure translational motion without any rotation. Its orientation remains horizontal throughout the motion.

Since the plate translates purely, every point on the plate, including its center of mass, moves in a circular arc of radius $L$.

Let the strings make a small angle $\theta$ with the vertical. The horizontal displacement of the center of mass is $x = L\sin\theta \approx L\theta$.

The restoring force acting on the plate is the horizontal component of the tension. Alternatively, considering the center of mass, the restoring force is:
$F = -Mg \sin\theta \approx -Mg\theta$

Substituting $\theta = \dfrac{x}{L}$, we get:
$F = -Mg \left(\dfrac{x}{L}\right)$

The acceleration of the plate is:
$a = \dfrac{F}{M} = -\left(\dfrac{g}{L}\right)x$

This represents simple harmonic motion with angular frequency $\omega^2 = \dfrac{g}{L}$.

Therefore, the time period of the motion is:
$T = \dfrac{2\pi}{\omega} = 2\pi \sqrt{\dfrac{L}{g}}$