# Archimedes` Principle

```Physics 2015: Archimedes’ Principle
Purpose
The purpose of today’s lab is to investigate Archimedes’
Principle and buoyant forces.
First, we will determine the density of metal balls of the
Then, using the known density of a metal slug, we will
determine the density of a liquid (water) and compare it
to the known density of water (an error calculation will be
required for this comparison)
Physics 2015: Archimedes’ Principle
Archimedes’ Principle
An object that is submerged in a liquid will experience an
additional upward force (due to the submersion) that is
equal to the weight of the amount of liquid that was
displaced by the object. This force is called the “buoyant
force”.
Translated into simpler language, this means:
An extra force acts on an object when it is placed, for
example, in water. This makes the object appear lighter
than it really is.
Physics 2015: Archimedes’ Principle
Example of a buoyant force
A metal ball of volume 50cm3 fully submerged in water
will experience a buoyant force of the following
magnitude:
mass of displaced water
density of water
Fb  mw g  w V g
(  0.001
volume of ball
kg
m
3

50
cm

9
.
8
 0.49 N )
3
2
cm
s
Physics 2015: Archimedes’ Principle
Activity 1: Finding the density of the metal spheres
• Force sensor
• Metal tray (place the metal
spheres into the tray)
• Glass cylinder filled with
water.
Physics 2015: Archimedes’ Principle
Free Body Diagram (3 forces)
string
density of water
T
Fb  mw g  w V g
W  mg  m V g
volume of ball
density of metal
Physics 2015: Archimedes’ Principle
Using Newton’s Second Law…

In this static situation there is no acceleration, therefore  F  0

T   w V g  m V g  0

T  V g ( m   w )
 By measuring T and V, and knowing the density of water and the
acceleration of gravity, the density of the metal can be calculated.
Physics 2015: Archimedes’ Principle
In Activity 1, we will use metal spheres in water:
Vsphere
4 3
4 3
  r  T   r g  m   w 
3
3
In Activity 2, we will use a cylindrical metal object in a fluid:
Vcylinder   r h  T   r h g  m   fluid 
2
2
Physics 2015: Archimedes’ Principle
There is one more idea you will want to consider
before starting Activity 1
• Remember to “Tare” your force sensor in a
clever way before any procedure. In this case,
first hang the empty basket on the force sensor
and into the water. THEN tare the force sensor.
This will negate both the weight of the basket
and the buoyant force on the basket in your
force measurement!
• Tared this way, when you graph the tension
versus r3 in Excel, you should expect the line to
go through the origin. Why do you think this is?
Physics 2015: Archimedes’ Principle
In Activity 1 (spheres)
4 3
V r
3

4 3
T   r g ( w  m )
3
Measure T for different size spheres (different radii) and determine
the density of the spheres.
measured
r
r
calculated
3
measured
T
T
plot
What is the slope?
Get m from slope.
r
3
Physics 2015: Archimedes’ Principle
Careful when doing the Excel Plot of T versus r3
Plotting T versus r3 and using m3 as your unit results in a
large number for the trendline slope.
When you display the trendline equation in the trendline
label, not all digits of the slope may be displayed. So, your
slope k result may be off by a factor of 10 or 100 etc.
Here is how to fix it:
Right click on the trendline label (not on the trendline).
Choose “Format Trendline Label”
Choose “Number” or
Choose “Scientific” with 3 or 4 digits.
Physics 2015: Archimedes’ Principle
Activity 2: Determining the Density of Water
from Buoyancy
•
Use the plastic cup (not the tall glass cylinder) and the
cylindrical metal mass (slug) for this activity. Fill the
plastic cup with approximately 250ml water (so it won’t
overflow when submerging the slug).
•
Attach the cylindrical metal mass to the end of the force
sensor with a string. To measure the buoyant force,
submerge the slug into the water
Cylindrical metal mass
Physics 2015: Archimedes’ Principle
Step 1: Hang slug
on force sensor
Step 2: Tare the
force sensor
Tare
Step 3: Hang slug into
Water - this way you
measure only Fbuoyant
Physics 2015: Archimedes’ Principle
Then get the density of the liquid from the
measured buoyant force using the equation:
Fb   water Vslug g
Note: You will need to measure the dimensions of the
slug to get its volume.
Physics 2015: Archimedes’ Principle
Determine the uncertainty in your calculated
density by estimating the uncertainties (errors)
in the buoyant force, and the dimensions of the
slug.
Fb  .....  ......
Diameter of slug : d  .....  .....
Height of slug : h  .....  .....
Physics 2015: Archimedes’ Principle
Remember (see homework 3) that the fractional
error is the error divided by the value.
Example:
Fb  1.4  0.2 N
0.2
Fractional error : FE Fb 
 0.14
1.4
Physics 2015: Archimedes’ Principle
Remember (see homework 3) that for
multiplications and divisions the fractional errors
Example:
0.5
a  1.5  0.5
 FEa 
 0.33
1.5
0.2
b  4.0  0.2
 FEb 
 0.05
4.0
0 .1
c  0.5  0.1
 FEc 
 0.20
0.5
a  b 1.5  4.0
d

 12
c
0.5
Physics 2015: Archimedes’ Principle
Fractional error in d :
FEd 
FEa   FEb   FEc 
FEd 
0.33  0.05  0.20
2
2
2
2
2
2
 0.39
Uncertainty in d  FEd * d  0.39 *12  4.7
Result : d  12  5
Physics 2015: Archimedes’ Principle
If there are power involved, the equations are
modified as follows:
Example:
0.5
 0.33
 FEa 
a  1.5  0.5
1.5
0.2
 0.05
 FEb 
b  4.0  0.2
4.0
0 .1
 0.20
 FEc 
c  0.5  0.1
0.5
a 2  b 1.52  4.0
 18

d
0.5
c
Physics 2015: Archimedes’ Principle
Fractional error in d :
FEd 
2 * FEa   FEb   FEc 
FEd 
2 * 0.33  0.05  0.20
2
2
2
2
2
2
 0.69
From the power of 2
Uncertainty in d  FEd * d  0.69 *18  12.4
Result : d  18  12
Physics 2015: Archimedes’ Principle
Activity 3: Determining the Density of Water
from Measurements of Volume and Mass
•
Use the plastic cup with its volume indicators for the
volume determination.
•
Use the electronic scale to measure the mass.
•
Estimate measurement uncertainties for volume and
mass.
•
Calculate the density and the uncertainty in the density
similarly to activity 2.
```
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