How can the natural frequency of a stationary cylinder be determined?

[Ry122]

In this problem I guess I would first need to model the position of the cylinder with an equation and from that solve for the natural frequency. But how do I go about doing this when the object's default position is stationary?

 

[gabbagabbahey]

What is the force on the cylinder when it is displaced from its equilibrium position by a distance $y$ (up or down)?

 

[Ry122]

It is equal to the weight of the water displaced I believe.
But how can I form an equation of motion from this?

 

[gabbagabbahey]

It is equal to the weight of the water displaced I believe.

There are two forces acting on the cylinder at any given time:

(1) The downward force of gravity

(2) The buoyancy force (pressure) exerted by the water

Find a mathematical expression for the net force on the cylinder when it is displaced from equilibrium by a distance $y$

But how can I form an equation of motion from this?

Newton's 2nd Law maybe?

 

[Nickie]

As a scientist, you are correct in approaching this problem by first modeling the position of the cylinder with an equation. This equation would be based on the physical properties of the cylinder, such as its mass, stiffness, and damping coefficient. From this equation, you can solve for the natural frequency of the system, which represents the frequency at which the cylinder will oscillate without any external forces acting on it.

To address the concern about the object's default position being stationary, it is important to note that even when an object appears to be stationary, it is still subject to various forces and can exhibit small oscillations around its equilibrium position. Therefore, the natural frequency can still be determined even if the object is not initially in motion.

To accurately model the position of the cylinder and determine its natural frequency, it may be helpful to conduct experiments or simulations to gather data on the object's physical properties. This data can then be used to create a mathematical model and solve for the natural frequency.

In summary, the frequency of oscillation can be determined by modeling the position of the cylinder and solving for the natural frequency using its physical properties. Even if the object appears to be stationary, it is still subject to forces and can exhibit small oscillations that can be accounted for in the mathematical model.