To find the frequency at which the thrust between the particle and the platform is zero at some point in the motion, we need to consider the conditions for this to occur.
Let's assume that the platform's vertical motion can be described by the equation:
y = A * sin(ωt)
Where: y is the displacement of the platform from its equilibrium position, A is the amplitude of the platform's motion, given as 10 cm (0.1 m) in this case, ω is the angular frequency of the platform's motion, t is the time.
The particle resting on the platform will experience an inertial force due to the platform's acceleration. This inertial force can be written as:
F_inertial = -m * ω^2 * y
Where: m is the mass of the particle, ω is the angular frequency of the platform's motion, y is the displacement of the platform.
For the thrust between the particle and the platform to be zero at some point, the inertial force acting on the particle must be canceled out by another force. In this case, we can consider the gravitational force acting on the particle:
F_gravity = m * g
Where: m is the mass of the particle, g is the acceleration due to gravity.
Setting the inertial force equal to the gravitational force:
-m * ω^2 * y = m * g
Simplifying and solving for ω:
ω^2 = g / y
ω = sqrt(g / y)
Substituting the given values: g = 9.8 m/s^2 (acceleration due to gravity) y = 0.1 m (amplitude of the platform's motion)
ω = sqrt(9.8 / 0.1) = sqrt(98) ≈ 9.90 rad/s
Therefore, the frequency at which the thrust between the particle and the platform is zero at some point in the motion is approximately 9.90 / (2π) ≈ 1.58 Hz.