A cylindrical can of diameter \(10.0 \mathrm{~cm}\) contains some ballast so that it floats vertically in water. The mass of can and ballast is \(800.0 \mathrm{~g}\), and the density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\) The can is lifted \(1.00 \mathrm{~cm}\) from its equilibrium position and released at \(t=0 .\) Find its vertical displacement from equilibrium as a function of time. Determine the period of the motion. Ignore the damping effect due to the viscosity of the water.

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. It's like a swinging pendulum or a mass attached to a spring that moves back and forth. The classic hallmark of simple harmonic motion is this repetitive, oscillating behavior.
In physics problems, simple harmonic motion can be analyzed using formulas that describe the displacement, velocity, and acceleration of an object in SHM. One key formula involves calculating the period (T) of the motion, which is the amount of time it takes to complete one full cycle of oscillation. This can be determined by the equation:
\[ T = 2 \pi \sqrt{\frac{m}{k}} \]
Here, 'm' represents the mass of the object in motion, and 'k' is the spring constant, which is a measure of the stiffness of the spring in spring-mass systems. The angular frequency (ω), given by \( \omega = \frac{2 \pi}{T} \), describes how many radians the object passes through per unit of time. For a cylinder oscillating vertically in water, as described in the exercise, we treat the buoyant force as a 'restoring force' much like the force exerted by a spring. When the can is displaced from its equilibrium position and released, it exhibits SHM, oscillating about this central point.

Buoyancy is the upward force exerted by a fluid, such as water, that opposes the weight of an immersed object. It's the reason why objects like ships, buoys, and even our cylindrical can in the exercise stay afloat. Physically, buoyancy is a consequence of fluid pressure differences: the pressure at the bottom of an object submerged in a fluid is greater than the pressure at the top, due to the weight of the fluid above.
To calculate the buoyant force (FB), we use Archimedes' principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object:
\[ F_B = V_{\text{submerged}} \times \rho_{\text{water}} \times g \]
Where \(V_{\text{submerged}}\) is the volume of the fluid displaced by the object, \(\rho_{\text{water}}\) is the density of the fluid, and \(g\) is the acceleration due to gravity. In the exercise provided, we apply Archimedes' principle to find the equilibrium condition and use the concept of buoyancy to analyze the motion of the cylindrical can in the water.

Oscillations refer to back and forth motion around a central point, called the equilibrium position. In mechanics, oscillatory motion is seen in various systems, such as pendulums, springs, and even atoms in molecules. The oscillation is a fundamental concept that applies to many areas of physics including acoustics, optics, and quantum mechanics.
In the context of our problem with the cylindrical can, when the can is displaced and released, it exhibits oscillatory motion by moving up and down in the water. This motion fits into oscillations in mechanics because it follows a predictable pattern governed by Newton’s laws. To fully analyze this behavior and predict the can's position at any given time, we invoke the principles of harmonic motion. The oscillation can be described by the displacement function:
\[ y(t) = Y_0 \cos(\omega t) \]
Here, \(y(t)\) is the vertical displacement at time \(t\), \(Y_0\) is the initial displacement (amplitude), and \(\omega\) is the angular frequency. Using these concepts, we can predict how the can will move over time, which is crucial for fields like engineering where understanding the dynamics of structures in fluid environments is necessary.