# Box floating in a liquid and undergoing simple harmonic motion

• lower8
In summary, the period of oscillation for a box floating in a liquid can be determined by considering the relationship between the restoring force and displacement, using the equation for simple harmonic motion and taking into account the extra buoyancy force caused by the displacement of the box. This can be represented by the equation a = -Pho*area(\Deltax)/m and d^{2}y/dt^{2}=-k/m*(\Deltax).
lower8

## Homework Statement

A box is floating in a liquid. It is pushed down then released to oscillate. How do I determine the period of this oscillation?

No idea where to go with this one.

## The Attempt at a Solution

What form of eqn for SHM usually appear in and on what is it based?

Here the restoring force is related to displacement: First consider the dipping portion of the diaplacement--what happens to buoyancy as a funtion of depth--put in terms of delta h; when bobbing up the distance between center of mass for neutral buoyancy and the actual Y displacement is now the PE, but instead of thinking of as mgy, it might be best to consider an "excess" of buoyancy. In other words, develop an equation that related forse and displacement--from SHO this and mass should lead to period--ie look for an eqn using T, k, m

Ok, so I have that the extra buoyancy force would be the density of the liquid times volume of the extra displacement times g. However, I'm not seeing where T or k would come in.

Here is what comes to mind:

Force= ma=-(pho)*area*(Xo+$$\Delta$$x)-mg where mg is the weight of the object.

Let Xo be the displacement leading to neutral buoyancy--ie -(pho)A*Xo + mg = 0

Then a = -Pho*area($$\Delta$$x)/m

d$$^{2}$$y/dt$$^{2}$$=-k/m*($$\Delta$$x)

Look familiar?

## 1. How does a box float in a liquid?

A box floats in a liquid due to the principle of buoyancy. The weight of the box is balanced by the upward force exerted by the liquid, known as the buoyant force.

## 2. What factors affect a box's motion while floating in a liquid?

The main factors that affect a box's motion while floating in a liquid are the density of the box, the density of the liquid, and the volume of the box. These factors determine the buoyant force and the weight of the box, which in turn affect the box's motion.

## 3. How is simple harmonic motion related to a box floating in a liquid?

A box floating in a liquid can undergo simple harmonic motion when it is displaced from its equilibrium position and experiences a restoring force. This motion is similar to a spring-mass system, where the spring provides the restoring force.

## 4. What is the period of a box's simple harmonic motion while floating in a liquid?

The period of a box's simple harmonic motion while floating in a liquid depends on the properties of the box and the liquid, such as their densities and the size of the box. It can be calculated using the equation T = 2π√(m/k), where T is the period, m is the mass of the box, and k is the spring constant equivalent for the system.

## 5. How can the amplitude of a box's simple harmonic motion be changed while floating in a liquid?

The amplitude of a box's simple harmonic motion while floating in a liquid can be changed by altering the initial displacement of the box or by changing the properties of the system, such as the density of the liquid. A larger initial displacement or a lower density of the liquid will result in a larger amplitude of motion.

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