A horizontal platform is executing simple harmonic motion in the vertical direction of frequency v. A block of mass $\mathrm{m}$ is placed on the platform. What is the maximum amplitude of the platform so that the block is not detached from it?
(A) $\dfrac{8}{4 \pi^{2} v^{2}}$
(B) $\dfrac{\mathrm{mg}}{4 \pi^{2} v^{2}}$
(C) $\dfrac{\mathrm{g}}{2 \pi^{2} \mathrm{v}^{2}}$
(D) $\dfrac{\mathrm{g}}{\pi^{2} \mathrm{v}^{2}}$

Hint: We can say that in physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Any time the speed of an object is changing, it has an acceleration. Angular acceleration is defined as the rate at which the angular velocity is changing. If the Ferris wheel speeds up at a constant rate, then we would say that the angular acceleration is constant.

Complete step by step answer:
We know that angular acceleration is the change in angular velocity divided by time, while tangential acceleration is the change in linear velocity divided by time. People sometimes forget that angular acceleration does not change with radius, but tangential acceleration does. Acceleration is a change in velocity, either in its magnitude that is speed or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the speed might be constant.
We know that the maximum acceleration of mass $\mathrm{m}$ is $-\omega^{2} \times \mathrm{a}$
where a is the amplitude of SHM and $\omega=2 \pi v$ is the angular
frequency.
So, the maximum amplitude of the platform so that the block is not detached is given as $-\omega^{2} \times \mathrm{a}=-\mathrm{g} \Rightarrow \mathrm{a}=\mathrm{g} / 4 \pi^{2} \mathrm{v}^{2}$

Hence option A is correct.

Note: We know that a force is a push or pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. When a force acts (pushes or pulls) on an object, it changes the object's speed or direction (in other words, makes it accelerate). The bigger the force, the more the object accelerates. When a force acts on an object, there's an equal force (called a reaction) acting in the opposite direction. A force involves an interaction between two or more objects, and it causes a push or pull between the objects. Good examples of opposing force include drag due to interaction with an air mass and the force due to friction between two objects.