Physics Laboratory Manual
L A B O R A T O R Y
m Loyd
18
Archimedes' Principle
OBJECTIVES
•
A p p l y Archimedes' p r i n c i p l e to measurements of specific gravity.
•
Determine the specific g r a v i t y of several metal objects that are denser than water, a cork that is
less dense than water, and a l i q u i d (alcohol).
EQUIPMENT
LIST
• Laboratory balance, calibrated masses, 1000 m L beaker, string, alcohol
• M e t a l cylinders, and cork or piece of w o o d (to serve as u n k n o w n s )
THEORY
If the mass of an object is distributed u n i f o r m l y , its density p is defined as the mass m of the object
d i v i d e d b y its v o l u m e V. I n equation f o r m this is
0
Po = m/V
(Eq. 1)
The SI units for density are k g / m . Other c o m m o n l y used units for density are the cgs system units g / c m .
The specific gravity is defined as the ratio of the density of an object to the density of water p . The
equation for the specific gravity is given b y
3
3
w
Specific gravity = SG = pjp
(Eq. 2)
w
Specific g r a v i t y is a dimensionless q u a n t i t y because i t is the ratio of t w o densities. The density of water
i n the cgs system is 1.000 g / c m , and densities i n that system are numerically equal to the specific gravity
SG. Water has a density of 1000 k g / m , and densities i n the SI system are equal to SG x 10 .
Archimedes' principle states that an object placed i n a f l u i d experiences an u p w a r d buoyant force
equal to the w e i g h t of f l u i d displaced b y the object. The p r i n c i p l e applies to both l i q u i d s and gases.
I n this laboratory, w e w i l l examine the application of the p r i n c i p l e t o l i q u i d s . A n object floats i f its
density is less than the density of the l i q u i d i n w h i c h i t is placed. I t sinks i n the l i q u i d u n t i l i t displaces
a w e i g h t of l i q u i d equal to its o w n weight. The object is i n e q u i l i b r i u m because its w e i g h t acts d o w n w a r d and a buoyant force acts u p w a r d . A n object w i t h density greater than the l i q u i d i n w h i c h i t is
placed w i l l sink to the b o t t o m of the l i q u i d . I t experiences an u p w a r d buoyant force equal to the w e i g h t
of the displaced f l u i d , b u t that is less than the w e i g h t of the object, and i t sinks.
3
3
3
T H O M S O N
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189
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Physics Laboratory Manual • Loyd
For objects w i t h a density greater than the density of water, Archimedes' p r i n c i p l e allows a simple
determination of the specific g r a v i t y of the object. Consider an object that is tied to a string and submerged i n water w i t h the string attached beneath the a r m of a laboratory balance as s h o w n i n Figure 18-1.
Also i n the figure is a free-body d i a g r a m of the forces acting on the submerged object. If B stands for
the buoyant force on the object, W is the w e i g h t of the object, and Wi is the tension i n the string, the
f o l l o w i n g is true at e q u i l i b r i u m
(Eq. 3)
B + W, = W
The q u a n t i t y Wi is the apparent w e i g h t read b y the laboratory balance i n Figure 18-1. By Archimedes'
principle, the buoyant force is the w e i g h t of displaced water. This can be w r i t t e n as
B = mg = pVg
w
w
=
w
(Eq. 4)
p Vg
w
0
w i t h m the mass of the displaced water, and V and V the volumes of the displaced water and the
object. The last step i n Equation 4 is true because the object sinks and displaces water equal to its v o l u m e ,
and therefore V = V . U s i n g Equations 3 and 4 w i t h W = mg and Wi =m-[g gives
w
w
W
0
0
Wi = W -B
W
= mg-
mg
mg- m g
W-W
1
x
pVg
w
= pVg
w
0
-
0
(Eq. 5)
p Vg
w
0
PoVog
pVg
0
SG = ^-
0
=
- {p V g
0
o
-
p V^
w
0
Pzv
m
(Eq. 6)
(Eq. 7)
m - mi
This laboratory w i l l use Equation 7 to determine the specific g r a v i t y of several metals w i t h densities
greater than that of water.
To use Archimedes' p r i n c i p l e to determine the specific g r a v i t y of an object that floats, the object
must be submerged b y attaching a lead w e i g h t to the object. Figure 18-2(a) shows the lead w e i g h t i n
water and the object i n air, and the balance reading is W . I n Figure 18-2(b) both the object and the
lead w e i g h t are below the water, and the balance reads W . By analysis similar to that used to derive
Equation 7, i t can be s h o w n that the specific g r a v i t y of an object that floats i n water is given by
a
2
SG
W
Wi
- W>
mg
m\g - m2g
ni
Beaker
(Eq. 8)
in
B
Tension in String is W,
Water
m
m\ m-i
o
w
Unknown
Figure 18-1 Determining the apparent mass of an object w i t h density greater than water.
Laboratory 18 • Archimedes' Principle
191
B
W
(a)
W
(b)
Figure 18-2 Forces acting on an object held submerged by a lead weight.
In Equation 8 m is the mass of the object, m is the apparent mass w i t h the object i n air and the lead
w e i g h t submerged i n water, and m is the apparent mass w i t h the object and the lead w e i g h t b o t h
submerged i n water.
1
2
The specific g r a v i t y of a l i q u i d can be determined b y measuring the mass of some object m, the
apparent mass of the object submerged i n water m , and the apparent mass of the object submerged i n
the l i q u i d m . I t is assumed that the object sinks i n both liquids. By similar analysis as used t o derive
Equation 7, i t can be s h o w n that the specific g r a v i t y of the l i q u i d is given b y
w
L
SG (liquid)
m - mi
m - m
(Eq.
w
EXPERIMENTAL
PROCEDURE
Object Density Greater than Water
1. Use the laboratory balance to determine the mass of each of the t w o u n k n o w n metal cylinders.
Record these values as m i n Data Table 1. The u n k n o w n s should have their metal type stamped o n
t h e m , or else ask y o u r instructor f o r the metal type of the u n k n o w n s . Record the metal of the
u n k n o w n s i n Calculations Table 1 .
2. Place a clamp on the laboratory table and screw a threaded r o d i n the clamp. Slide the r o d into the
hole designed for that purpose i n the base of the laboratory balance. A d j u s t the height of the balance
above the table to a l l o w the 1000 m L beaker to f i t under the balance. Fill the beaker about three-fourths
f u l l of water.
3. Tie a piece of light string or thread a r o u n d one of the u n k n o w n s and suspend it f r o m one
i n the left a r m underneath the balance. Determine the apparent mass w i t h the u n k n o w n
completely below the surface of the water. Be sure that the u n k n o w n is not touching the
beaker or is i n any w a y supported b y a n y t h i n g other than the string. Record the mass
the balance as m^ i n Data Table 1 .
of the slots
suspended
side of the
reading of
4. Repeat Steps 1 t h r o u g h 3 for the second u n k n o w n .
5. F r o m A p p e n d i x I I obtain the k n o w n SG of the metal for each of the u n k n o w n s and record those
values i n Data Table 1 .
Object Density Less than Water
1. U s i n g the laboratory balance, determine the mass of the cork and record it as m i n Data Table 2.
2. Determine the apparent mass w i t h the cork suspended i n air and a lead w e i g h t tied below submerged
i n water as s h o w n i n Figure 18-2(a). Record that value as mi i n Data Table 2.
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Physics Laboratory Manual • Loyd
3. Determine the apparent mass w i t h b o t h the cork and the lead w e i g h t submerged i n water as s h o w n
i n Figure 18-2(b). Record that value as m i n Data Table 2.
2
Liquid Unknown
1. Use one of the u n k n o w n metal cylinders f r o m the original procedure. Its mass m and its apparent
mass i n water have already been determined. Record those values i n Data Table 3.
2. Use a very clean beaker and f i l l i t about three-fourths f u l l of alcohol. Determine the apparent mass of
the metal cylinder i n alcohol. Record i t as m i n Data Table 3. Return the alcohol to its container
w h e n y o u have finished w i t h i t . Be v e r y careful not to have any spark or flame near the alcohol.
L
3. Repeat Steps 1 and 2 for the other u n k n o w n metal.
4. F r o m A p p e n d i x I I determine the k n o w n value of the specific g r a v i t y of alcohol and record i t i n Data
Table 3.
CALCULATIONS
Density Greater than Water
1. Use Equation 7 to calculate the SG for each of the u n k n o w n s . Record the experimental values of the
specific gravity i n Calculations Table 1.
2. Calculate the percentage error i n y o u r value of the SG for each of the u n k n o w n s compared to the
k n o w n values. Record t h e m i n Calculations Table 1.
Density Less than Water
1. Use Equation 8 to calculate the specific gravity SG for the cork a n d record i t i n Calculations Table 2.
Liquid Unknown
1. Use Equation 9 to determine the specific gravity of the alcohol for the t w o trials w i t h the different
metals. Record those values i n Calculations Table 3.
2. Calculate the percentage error i n each of the t w o measurements of the specific g r a v i t y of alcohol.
Record the results i n Calculations Table 3.
Name
Date
Section
Lafc Partners
L A B O R A T O R Y
LABORATORY
18
Archimedes' Principle
REPORT
Data Table 1
Metal
Calculations Table 1
Known SG
m (kg)
mi (kg)
Data Table 2
m
m—m\
% Error
Calculations Table 2
m (kg)
m (kg)
m (kg)
1
2
Data Table 3
Metal
SG =
SG
m
m\—m2
Calculations Table 3
m (kg)
m (kg)
w
m (kg)
L
SG =
~
m-m
m
m
L
% Error
w
K n o w n SG of Alcohol =
x
O
195