a horizontal plate is vibrating vertically with simple harmonic motion at a frequency of 20 Hz.What is the amplitude of vibration so that the fine sand on the plate always remain in contact with it?
To determine the amplitude of vibration required for the fine sand to always remain in contact with the plate, we need to understand the concept of simple harmonic motion.
Simple harmonic motion (SHM) refers to the oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium point and acts in the opposite direction. In the case of a vibrating plate, the sand particles will experience an upward force when they are below the equilibrium point, causing them to remain in contact with the plate.
The frequency of the vibrating plate is given as 20 Hz, which represents the number of complete oscillations per second. We can determine the angular frequency (ω) using the formula:
ω = 2πf
where f is the frequency and π is approximately 3.14159.
ω = 2π * 20 Hz = 40π rad/s
Now, let's assume that the amplitude of the vibrating plate is A meters. The displacement of the vibrating plate can be represented by the equation:
y(t) = A * sin(ωt)
where y(t) represents the displacement of the plate at a given time t. For the sand particles to always remain in contact with the plate, the displacement of the plate should be equal to or greater than the thickness of the sand layer.
Since the sand remains in contact with the plate at both the highest and lowest points of the vibration (the amplitude), the amplitude should at least be equal to the thickness of the sand layer. Therefore, the required amplitude (A) is equal to the thickness of the sand layer.
To determine the exact value of the amplitude, you would need to measure the thickness of the sand layer on the plate.