Final answer:
The largest displacement amplitude that a table can have while still keeping an object in contact is when the maximum acceleration of the table (determined by the square of the angular frequency and the amplitude) does not exceed the acceleration due to gravity. This ensures that the object will not lose contact because of inertial forces at the point of maximum vertical acceleration.
Step-by-step explanation:
The question asks about the largest displacement amplitude (maximum amplitude, X) that a table with vertical sinusoidal motion can have, ensuring that an object on the table remains in contact with it at all times. The amplitude in simple harmonic motion is the maximum displacement from the equilibrium position. The preservation of contact between the table and the object will depend on the maximum acceleration that the motion can impart on the object without overcoming the gravitational force acting on it. The moment when the acceleration due to the sinusoidal motion is equal to the gravitational acceleration is the critical point. Beyond this, the object will lose contact with the table.
Since the acceleration in sinusoidal motion (simple harmonic motion) is proportional to the displacement and, more specifically, in this context, the table's maximum acceleration is at the maximum displacement (amplitude). We can understand that to keep the object in contact with the table, the product of the angular frequency squared (ω²) and the amplitude (X) must not exceed the acceleration due to gravity (g). The formula for maximum acceleration in simple harmonic motion is Amax = ω²X. With gravity being g, the condition is ω²X <= g.
In conclusion, the maximum amplitude is determined by the relationship between the square of the angular frequency of the table's motion and the acceleration due to gravity. The largest possible amplitude is when ω²X equals g. This ensures that, even at the table's highest or lowest point, where the acceleration is greatest, it does not exceed g and allows the object to remain in contact without losing contact due to inertial forces.