A vibration platform oscillates up and down with an amplitude of 8.8 cm at a controlled variable frequency. Suppose a small rock of unknown mass is placed on the platform. At what frequency will the rock just begin to leave the surface so that it starts to clatter?
To determine the frequency at which the rock will just begin to leave the surface of the vibrating platform, you need to consider the gravitational force acting on the rock and the maximum acceleration experienced by the rock on the platform.
1. Determine the maximum acceleration experienced by the rock:
- The amplitude of oscillation is given as 8.8 cm. Convert this to meters by dividing by 100: 8.8 cm = 0.088 m.
- The maximum acceleration (a_max) experienced by an object in simple harmonic motion is given by the equation a_max = ω^2 * A, where ω is the angular frequency and A is the amplitude.
- In this case, A = 0.088 m.
- Since no information about ω is given, you need to find it using the following formula:
ω = 2πf, where f is the frequency in Hz.
- Substitute the values: a_max = (2πf)^2 * 0.088.
2. Determine the gravitational force acting on the rock:
- The gravitational force can be calculated using F_gravity = m * g, where m is the mass of the rock and g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Since the mass of the rock is unknown, we'll refer to it as m.
3. Equate the gravitational force and the maximum acceleration:
- Set F_gravity = a_max and solve for the frequency (f) at which the rock just begins to leave the surface of the platform.
- Substitute the values: m * g = (2πf)^2 * 0.088.
With this equation, you can now solve for the frequency (f). Keep in mind that the mass of the rock (m) is unknown, so you won't be able to calculate an exact value for f. However, you can find a range of possible values once you know the mass of the rock.