CK-12-Physics - Intermediate

http://www.ck12.org Chapter 12. Fluid Mechanics

12.4 Archimedes’ Law

Objectives

The student will:

• Understand buoyancy and how it applies to Archimedes’ Law.


• Be able to solve problems using Archimedes’ Law.

Vocabulary

• buoyancy: The force that pushes upward on an object that is partially or wholly submerged in water.

Introduction

Objects float in water or sink based on their density. Buoyancyis the force that pushes upward on an object that is
partially or wholly submerged in water. This happens because the water lower down has higher pressure than the
water around the top, so the bottom of the object is pushed up more than the top of the object is pushed down.

For an object of irregular shape, calculating the force as pressure times area,F=P×A, could potentially be
difficult. However, there is a simple principle for how much force comes from buoyancy, attributed to the ancient
Greek engineer and physicist Archimedes (287 BC –212 BC). Archimedes’ principle states the following:

The force of buoyancy on an object immersed in liquid is equal to the weight of liquid displaced by that object.

Expressed mathematically, the buoyant forceFBis equal to the weight of the liquid displaced, and we know that
weight ismg. The mass of the liquid displaced is equal to mass times density,ρV. This gives us:

FB=mg= (ρliqV)g

This is phrased simply, but can be shown to have many immediate consequences.

• If an object is more dense than water, then water with equal volume to it has less mass. If so,FBis less than
  the object’s weight, and it will sink.


• If an object is less dense than water, then water with equal volume to it has more mass. If so,FBis more than
  the object’s weight, and it will float.

We can derive this by looking at a case of a simple cube floating upright in the water. The pressure of the water is
alwaysP=ρgh, wherehis the depth in the water. The cube has areaAfor all faces. It is pushed down by the
pressure on the top,Ftop=PtopA, while is it pushed up by the higher pressure on the bottom,Fbottom=PbottomA. The
difference between these is: