To determine the amplitude of vibration required for the fine sand to always remain in contact with the vibrating plate, we need to consider the critical condition at the maximum displacement.
Let's denote the amplitude of vibration as A, the frequency as f, and the acceleration due to gravity as g.
At the maximum displacement, the upward acceleration due to the plate's vibration must be equal to the downward acceleration due to gravity for the sand to stay in contact with the plate.
The upward acceleration at any position x can be determined using the equation for simple harmonic motion:
a_up = -ω²x
where ω is the angular frequency given by ω = 2πf.
The downward acceleration due to gravity is simply -g.
At maximum displacement, when x = A, the upward acceleration is a_up = -ω²A.
Setting this equal to the downward acceleration due to gravity, we have:
-ω²A = -g
Substituting ω = 2πf into the equation, we get:
-(2πf)²A = -g
Simplifying the equation, we find:
4π²f²A = g
Finally, we can solve for the amplitude A:
A = g / (4π²f²)
Substituting the given values, g ≈ 9.8 m/s² and f = 20 Hz, we get:
A = (9.8 m/s²) / (4π² * (20 Hz)²)
Evaluating this expression, we find A ≈ 6.21 × 10^(-4) meters.
Therefore, the amplitude of vibration required for the fine sand to always remain in contact with the plate is approximately 6.21 × 10^(-4) meters.