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## Resources

Modeling Scenario

## 3-055-FloatingBox-ModelingScenario

Author(s): John Thoo

Keywords: oscillation Archimedesâ€™ Principle buoancy oscillatory motion Newton's Second Law

## Abstract

In this scenario, we lead students through the process of building a mathematical model for a floating rectangular box that is bobbing up and down.

## Citation

Researchers should cite this work as follows:

## Description

The principal forces acting on the box are its weight (the force due to gravity) and the buoyancy force (that makes it float). By Archimedes' Principle, the buoyancy force acting on an object equals the weight of the liquid that is displaced by the object.

If an object that is in water is in equilibrium (neither sinking nor rising), then the buoyancy force that is acting on the object is equal in magnitude and in the opposite direction to the force due to the weight of the object. If the object is submerged below its equilibrium (floating) position, then the buoyancy force will cause the object to rise. The upward motion would move the object above its equilibrium position, and then the weight of the object would cause it to fall. The downward motion would move the object below its equilibrium position, and then the buoyancy force would cause the object to rise again, and the cycle would repeat indefinitely (bobbing up and down) if there were no friction (damping, drag).

Activity 1

The goal is to come up with a mathematical model for a floating rectangular box that bobs up and down in water.

Activity 2

Suppose that the box described above develops a leak in its bottom, and that water is flowing into the box.

Activity 3

In Activity 1, we assumed that the box is floating in still water, and that the only forces that are acting on the box are its weight and the buoyancy. We now add the assumption that a third force, friction or drag, is also acting on the box.

Activity 4

The box is held with its bottom 0.8 m below the surface and then released from rest.

## Authors

Author(s): John Thoo