density

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PHYSICS DensityMr. Omosa Elijah
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DENSITY It is often useful to know not simply the
mass of an object but rather the mass of a
definite amount of the material of which it is
made. The mass of a unit volume of a substance is
called the DENSITY of the substance and is
measured in kilogram per cubic metre (kg/m3) or
gram per cubic centimetre (g/cm3).
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Density Amount of mass per unit volume of a
substance.

• SI Units kg/m3
• Common Units g/cm3 or g/mL

Problem Drunken Donny steals an unknown alcohol
from the chemistry lab at work. He does not know
that there are numerous different types of
alcohols. Methyl alcohol has a density of 0.792
g/mL and is poisonous if consumed. Ethyl alcohol
has a density of 0.772 g/mL and is the common
alcohol which Drunken Donny loves to drink. If
the stolen unknown alcohol has a measured mass of
71.28 g and a measured volume of 90.0 mL, which
alcohol did Drunken Donny steal to drink?
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Density is how much matter is in something
(mass), compared to the amount of space it takes
up (volume).
The formula for density is
Mass (grams)
divided by
Volume (cm3)
So the unit for density is g / cm3

• Every substance has a density, and that density
  always remains the same.
• Density can be used to figure out what an
  unknown substance is.
• The density of water is 1 g / cm3

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Examples 1. What is the density of a metal if 4
m3 of it have a mass of 28 000 kg? density
28000/4 7000kg/m3. 2. What is the mass of 0.5
m3 of copper? mass density x volume 8930 x 0.5
4465 kg. 3. What is the volume of 9000 kg
(about 9 tonnes) of concrete, density 3000
kg/m3 Volume mass/density 9000/3000 3m3.
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Density of Irregular shaped objects Take one of
the irregular shaped objects and use the balance
to measure its mass in grams. Put some water into
the measuring jug up to a known level. Tie it to
a piece of cotton to the object and lower it into
the water so that it is completely below the
surface. See how much the water level rises
this is the volume of the object. Write down your
results in the table.
Material Mass (gms) Volume (cm3) Density (g/cm3)





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• Example You can find the volume of an irregular
  solid by putting it in water in a measuring
  cylinder and seeing how much the water level
  rises. The object shown in
• Figure 1 has a mass of 200 g.
• (a) What is its volume?
• What is its density?
• Soln
• V V2 V1 ? m/v
• V 75 60 200/15
• V 15cm3 13.3g/cm3

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Example KNEC Figure 3 shows a metal cube of mass
1.75g placed between the jaws of a micrometer
screw gauge. The magnified portion of the scale
is also shown. The reading on the gauge when the
jaws were fully closed without the cube was 0.012
cm. Use it to answer questions 5 and 6.

1. What is the length of the cube? ( 1 mark)
2. 6 Determine the density of the metal cube giving
   your answer correct to 3 s.f.s. ( 3 marks)

Figure 3
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• Example KNEC
• What is the length of the cube?( 1 mark)
• 0.550 cm 0.562 0.012
• 5.62 0.12 5.50mm.
• Determine the density of the metal cube giving
  your answer correct to 3 s.f.s.( 3 marks)
•  Density p m/v
• 1.75g 10.5cm3
• (0.550)3 cm3

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Relative density The density of a substance is
often compared with that of water and this is
called the RELATIVE DENSITY of the substance.
Relative density mass of substance/mass of an
equal volume of water
On this scale, iron would have a relative density
of 7.87, methylated spirit 0.79 etc.
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• Density and Density Bottles
• The figure below show density bottle with the
  masses as recorded when empty, when filled with
  water and when filled with oil.

Empty bottle, 20g, Bottle filled with
water, 50g, Bottle filled with oil,
45g If the density of water is 1 g/cm3,
determine the relative density of the oil. Hence
determine its density
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Density and Density Bottles contd. Soln Results
and Conclusions Mass of density bottle M0 Mass
of density bottle with water M1 Mass of the
density bottle with oil M2 Relative density of
liquid A Mass of oil
Mass of
equal. Vol. water M2 - M0
M1 - M0 45 - 20 50 20
25/30 x 1
0.83g/cm3
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• Problem 2
• In determining the density of granules, a student
  obtained the results as shown below.

Empty bottle, 20g Bottle with some granules
Bottle granules water Bottle filled with
35g 65 g water
only, 50g Determine the density of the
granules in kg/ m3. Relative density of liquid A
Mass of granules
Mass of equal. Vol.
water M2 - M0 35 - 20
M3 M4 65 50 15/15 x 1000
1000 kg/cm3
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Upthrust and Archimedes principle

• A weight feels much lighter to lift in water
  than outside the water.
• He said that the water gives an upward force or
  upthrust on any object in it.

If you already know the density of the liquid
then you can simply measure the volume of
displaced water and use mass volume x density
to find its mass. This can easily be done using a
measuring cylinder as shown in the diagram.
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Upthrust and Archimedes principle

• You can see how the apparent weight of the stone
  gets less when it is immersed in water. If it was
  only partly immersed it would appear to weigh
  less than in air but not as little as when it is
  totally immersed in the water.
• You can weigh an object in air and then in water
  and actually work out the upthrust, it is the
  difference between the two readings.

Upthrust apparent loss of weight of object
weight in air - weight in liquid.
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Upthrust and Archimedes principle The weight of
liquid displaced is the weight of the liquid that
has been replaced by the object. The volume of
this amount of liquid is equal to the volume of
the object itself. The weight of fluid displaced
and therefore the upthrust will be bigger if the
density of the liquid is large.
The upthrust in salty water (relative density
1.1) is larger than that in water (relative
density 1.0) for the same object. This is why
it is easier to swim in the sea than in a
freshwater lake.
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Upthrust and Archimedes principle

Archimedes principle states that
When a body is partly or totally immersed in a
fluid there is an upthrust that is equal to the
weight of fluid displaced.
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EXPERIMENT 2 Weigh an object in air and then
lower it into a beaker of water that is resting
on a top pan balance. The reading on the spring
balance will get less while the reading on the
top pan balance will increase by the same amount.
This is true for all liquids
Volume of displaced water volume of stone
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Example Determine the density of glass that
weighs 0.5N in air and 0.3N in water. Solution Rel
ative density weight in air loss of wt.
in water Relative density 0.5
0.5 -0.3 density 0.5/0.2 x 1000 2500
kg/m3
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Proof of Archimedes principle Consider a uniform
cylinder immersed in a liquid as shown in Figure
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Force on the upper face of the cylinder
hrgA Force on the lower face of the cylinder h
LrgA Difference in force LrgA But LA is the
volume of liquid displaced by the cylinder, and
LrgA is the weight of the liquid displaced by the
cylinder.
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Proof of Archimedes principle contd. Therefore
there is a net upward force on the cylinder equal
to the weight of the fluid displaced by it. The
same result will be obtained for a body of any
shape, regular or not by taking into account the
vertical and horizontal components of the forces
on the object. If a sphere of radius r made of
material of density r is fully immersed in a
liquid of density s the apparent weight of the
sphere is given by
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Proof of Archimedes principle contd. Apparent
weight actual weight - upthrust 4/3 pr3g(r -
s) The fluid may be a liquid, such as water, or a
gas, such as air, although due to the low density
of air (about 1 kg m3) the upthrust in air in
usually small, but sufficient to support a
helium-filled balloon or a hot air balloon.
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Example A lump of iron of mass 8 kg is hung in
brine of density 1100 kg/m3. If the iron has a
volume of 0.001 m3, find a) the density of
iron b) the loss of weight in brine c) the
apparent weight in brine. solution a) density
8/0.001 8000 g/cm3 b) loss of weight
0.001x 1100 1.1x10 11 N c) apparent weight
80 11 69 N (Force of Earths gravity (g)
10 N/kg)

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• Examples
• A 20 kg spherical hollow steel buoy of volume
  0.06 m3 is tethered to the bottom of a fast
  flowing river by a cable so that the cable makes
  an angle of 400 with the base of the river.
  Calculate the tension (T) in this cable.
• Resolving vertically and taking g 9.8 ms-2
• Forces on the buoy
• Upthrust mgTcos40 20g Tcos40But
  Upthrust 0.06 x 1000 x g
• Therefore 20g Tcos40 0.06 x 1000 x g so
  
• T cos 40 60g 20g 40gT 511 N

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Examples 2. A hot air balloon with a volume of
200 m3 hangs in the air. If the density of the
hot air is 0.8 kgm-3 and that of the cool air
outside the balloon is 1.2 kgm-3 what is the
biggest load it can support if the fabric of the
balloon and the basket have a total mass of 60
kg. (Take g 9.8 ms-2) Weight of balloon and
basket 60 g 588 N Upthrust Weight of air
displaced 200x1.2x g 2352 N Weight of hot air
in the balloon 200x0.8x9.8 1568 N Total
weight of balloon and hot air 1568 588 2156
N Therefore additional load that can be supported
by the balloon 2352 2156 196 N
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KNEC Question A piece of marble of mass 1.4kg
and relative density 2.8 is supported by a light
string from a spring balance. It is then lowered
into the water fully. Determine the up
thrust. (i) Lower surface of solid P ?gh Force
P x A gh x A 800 x 10 x 0.5 x 4 x 10-4
0.96N Upper surface of solid. Force P x A
? gh x A 800 x 10 x 0.1 x 4 x 10-4
0.32N (ii) Upthrust 0.96 0.32 0.6N Weight
of the solid Density x Volume x g 2.7 x
103 x 0.2 x 104 x 4 x 10 2.16 Balance reading
2.16 0.64 1.52N
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KNEC Questions 1. The ball B shown below has a
mass of 12kg and a volume of 50litres. It is held
in position in sea water of density 1040 kgm-3 by
a light cable fixed to the bottom so that 4/5 of
its volume is below the surface determine the
tension in the cable. 2. A balloon of
volume 1.2x107 cm3 is filled with hydrogen gas of
density 9.0 x 10-5/g/cm3. Determine the weight
of the fabric of the balloon.
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Floating objects
It should be clear from the above that a floating
body will displace its own weight of fluid such
that there is no vertical resultant force on the
body. The volume of the floating object that is
below the surface will depend on both the density
of the object and that of the fluid in which it
is floating.
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Ferries
You can use Archimedes principle to get a rough
idea of the draught of a ship such as a ferry.
Example Mass 24 500 metric tons 24.5 x106
kg Upthrust weight of water displaced
24.5x106g N Density of sea water 1030
kgm-3 Volume of water displaced 24.5 x106/1030
2.38x104 m3 Length of ferry (assume a
rectangular section) 151 m Width of ferry 26
m Volume of ferry below the waterline Area x
draught 151x26xdraught 2.38x104 Therefore
draught 2.38x104/151x26 6.06 m (The actual
draught of the ferry is 6.2 m, slightly different
since we have assumed a rectangular cross
section)
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HYDROMETER A hydrometer is used to measure the
density of a liquid. This is a glass tube with
a scale on the side weighted at the bottom with
lead shot. The hydrometer floats in the liquid
and if the liquid is dense the hydrometer does
not need to sink very low to displace its own
weight of liquid. If the liquid has a lower
density it will sink much deeper. Hydrometers are
used to check the density of car battery acid,
beer, milk and wine. Two types are drawn below.

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KNEC Qn Figure 1 shows a block with a graduated
side, and of dimension 4cm x 4cm x 16cm, just
about to be lowered into a liquid contained in an
overflow can.
During an experiment with this set-up, the
following information was recorded -The block
floated with three quarters of it
submerged -Initial reading of balance 0
g -Final reading of balance 154 g

• Use the information to determine the density of
  the
• Block (4marks)
• Liquid ( 3marks)
• (Use g 10ms-2 . give your answers to 1 decimal
  place.)

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• KNEC Qn
• (a) State the law of floatation. ( 1 mark)
• Figure 13 shows a simple hydrometer
• (i) State the purpose of the lead shots in the
  glass bulb
• (1 mark)
• (ii)How would the hydrometer be made more
• sensitive? ( 1 mark)
• (iii) Describe how the hydrometer is
• calibrated to measure relative density (2
  marks)

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