Density
Density of an object is the ratio of
the mass of the object to its volume. It is a salar quantity. It is measured in
kilogram per metre cube (kg/m cube) or gram per centimetre cube (g/cm cube).
Formula
for Calculating Density
Since density is the ratio of mass of
object to its voume, the formula for calculating density is as stated below:
Density
= mass of object m / volume of object, v
Density
= m/v
Also,
Density
= mass / l * b * h
Or
Density
= mass / (area of object * height of object)
Density
= mass / (area of object * thickness of object)
Where,
L
= length; b = breadth; h = height (of object)
Measurement
of Density of a Substance
The density of a substance can be
measured or determine by dividing the mass of the substance by the volume of
the substance.
The mass of the substance is obtained
by using spring balance to measure its mass while its volume is determined by
calculation using any suitable formula. For an irregular object, the method
explained below is used to determine its volume by law archimedes’ principle.
Determination
of Volume of Irregular Objects
Irregular objects are objects that do
not have a definite shape. For such objects, there is no suitable formula to
calculate their volumes. The only way to determine their volume is by the
method explains below.
Aim:
To determine the volume of irregular
object.
Apparatus:
Overflow can, irregular object, water,
thread, measuring cylinder, beaker.
Set-up
Diagram:
Procedures:
Fill water in an overflow can. Tie the
irregular object to a thread and immerse the object into the water in the
overflow can.
Use the beaker and collect water
displaced by the object immersed in the water.
Use the measuring cylinder and measure
the volume of the displaced water.
Conclusion:
The volume of the water displaced is
the volume of the irregular object.
At this point, you use the determined
volume of the irregular object and its mass to calculate the density of the
object.
Experiment
to Determine the Density of a Liquid
Aim:
To determine the density of a liquid.
Apparatus:
Density bottle, liquid, chemical
balance,
Set-up
Diagram:
Procedure:
Weigh a clean and dry density bottle
and record its mass as M1
Fill the density bottle with water, clean
it, reweigh and record its mass as M2
Empty the bottle, fill it with a
liquid such as oil or paraffin.
Weigh and record its mass as M3.
Data
Obtained
Mass
of empty bottle = M0
Mass of botlle and water = M1
Mass of bottle and oil = M2
Mass of water = M1 - M0
Mass of oil = M2
– M0
Calculation:
Formula:
Density of liquid = mass of liquid /
mass of equal volume water
Substitution:
Density
of liquid = (M2 – M0)/(M1 –
M0)
Experiment
to Determine the Density of a Liquid
Aim:
To determine the density of air.
Apparatus:
50 litres plastic container with a
tap, air pump, air,
Set-up
Diagram:
Procedure:
·
Open
the tap of the 50 litres container.
·
Weigh
the 50 litres plastic container that contains air at atmospheric pressure and
record its mass as M0.
·
Pump
air into the container until it is very hard and close the tap.
·
Reweigh
and record the mass of the container and record as M1.
·
If
there is no increase in mass, pump in more air.
·
By
means of a tube connected to the tap, release the air from the container and
collect it over water in a transparent glass (of cross sectional area 100cm²)
whose open end is placed under the water.
·
Make
a line L on the glass at a distance of 10cm, so that a volume of 1000cm³ or
1Litre is defined.
·
Adjust
the container so that the water level inside the glass and outside are in line
with the mark line L. at this level, 1Litre of air at atmospheric pressure has
been collected.
·
Close
the tap of the container. Refill vessel B with water and repeat the experiment.
·
Reweigh
the 50 Litres container to know the mass of air collected and its volume and
record it as M2.
·
Let
volume of air collected = l
Data:
Mass of container A + pumped air = M1
Mass
of container A after air is removed =
M2
Mass
of air =
M1 -
M2
Calculation:
Density of air at atmospheric pressure = mass of air /
volume of air
Density
of air (atmp) = Massair / Volumeair
Substitution:
Density
of air (atmp) = M1
- M2) /Volumeair
Density
of Some Substance
The density of some substances are stated
beow:
Substance density in g/cm³ density
in kg/m³
Water 1g/cm³ 1000kg/m³
Mercury 13.6g/cm³ 13600kg/m³
Alcohol 0.79g/cm³ 790kg/m³
Turpentine 0.87g/cm³ 870kg/m³
Paraffin 0.79g/m³ 790kg/m³
Petrol 0.
Kerosene
Diesel
Oil
Application
of Formula
Worked
Examples:
1. An object of mass 24kg has a volume of
0,37m cube. Determine the density of the object.
Solution:
Data
given in the question:
Mass
= 24kg; volume = 0.37; density = ?
Formula:
Density
= mass / volume
Substitution:
Density = 24 / 0.37
Density
= 64.86kg/m cube
2. The density of a 2400g mass is 0.643
kg/m cube. What is the volume of the
object? Calculate the area of the object if its thickness is 0.25m.
Solution:
Data given in the question:
Mass
= 2400g = 2400/1000 = 2.4kg; density = 0.643kg/m cube.
Thickness = 0.25m
Formula:
Density
= mass / volume
Density
= mass / area * thickness
Substitution:
0.643 = 2.4kg / area*0.25
You
cross multiply and make area he subject of the formula.
Therefore,
0.643
* area*0.25 = 2.4
Area = (2.4) / (0.643*0.25)
Area = 14.93m sqr.
Relative
Density of a Substance
Relative density of a substance is the
ratio of the density of the substance to the density of equal volume of water.
It can also be defined as the number
of times that the density of the substance is greater or lesser than that of
water.
Relative density has no unit because
it is the ratio of the same units. It is scalar quantity.
Also, the relative density of a
substance is the number of times that 1cm³ of the substance is heavier than
1cm³ of water.
For example, the relative density of
lead is 11.4. This means that 1cm³ of lead is 11.4 times heavier than 1cm³ of
water.
At this point, I want you to note
that, the density of a substance in g/cm³ gives or becomes the relative density
of the substance. Therefore, since the density of lead is 11.4g/cm³, its
relative density is 11.4. This is applicable to other substances.
Also, you must know that the density
of a substance in kg/m³ does not give the relative density of the substance.
The reason is, the density of water in kg/m³ is 1000kg/m³ and not 1kg/m³. Therefore,
if you divide the density of the substance in kg/m³ by the density of water in
kg/m³ (1000kg/m³), you will get a different value that is different from the
value of the density of the substance in kg/m³
For example, if the density of a
substance is 23.5kg/m³, if you divide it by the density of water which is 1000kg/m³
(23.5/1000=0.0235), it will not give you 23.5kg/m³ rather it will give you
0.0235 as the relative density of the substance. Therefore, the density of a
substance in kg/m³ does not give the relative density of the substance.
Formula:
The formula for calculating the
relative density of a substance is as follows:
Relative
density = density of substance / density of water
Relative
density = Densitysubstance / Densitywater
Or
Relative
density = mass of substance / mass of equal volume of water
Or,
Relative density = mass of any volume
of substance / mass of equal volume of water
Or
You can state relative density as
ratio of weights.
Therefore:
Relative density = weight
of substance / weight of equal volume of water
Relative density = weight
of substance / upthrust of water
Floating
objects
Balloons:
Balloon that I filled with air or gas
floats. When a balloon that contains gas is released, the balloon rises up due
to upthrust of air on it. The balloon rises to a point where upthrust of the
air is equal to the weight of the balloon and the gas in it. At this point,
the balloon floats and move about in the air.
.
Cartesian diver
Cartesian diver is a small hollow
figure made of glass, open-ended tail. When it is full of air, it floats on
water. If the diver is put inside a bottle full of water with a fitting cork,
it can be made to sink by pressing in the cork. The pressure on the cork
increases the pressure inside the bottle and forces water to enter inside the
body of the diver through the tail. The diver’s body weight is now consist of
glass, water and air. The diver will sink or remain stationary or rise in the
bottle depending on whether its total weight is greater than or equal to or
less than the weight of water displaced.
Submarine
A submarine has a ballast tank which
contain water. The quantity of water in the ballast tank determine the buoyancy
of the submarine. When the submarine is required to dive, water is allowed into
the ballast tank. When the submarine wants to rise and float, water is ejected
from the tank by a compressed air.
Ice
melting in water
Supposing an ice of mass 9g floats in
water in a container, the volume of the water displaced by the ice is 9cm³,
according to law of flotation. If the density of the ice is 0.9g/cm³, then the
volume of ice is 10cm³. the remaining 1cm³ is the portion of the ice that is
above the water surface.
When the ice melts, there will be no
increase in the volume of the liquid. The melted ice would occupy take the
space of the meted ice and the water level would not increase.
Hydrometer
Hydrometer is an instrument that is
used to determine the relative density of a liquid.
It consists of a calibrated tube of
two diameters. The end of the lower tube contains lead shots which enable the
instrument to sink to a certain depth in a liquid. It is used to measure the relative density of
acid in car battery. When the battery is fully charged, relative density is
1.15 more than that of water.
Hydrometer
diagram here
Determination
of solid and liquid by a balance lever
1. Relative
Density of Solid
ARCHIMEDES
PRINCIPLE
Archimedes principle explains what
happen when an object is partially or totally immersed in a liquid (fluid).
When an object is partially or totally
immersed in a liquid, the object experiences an upthrust and becomes lighter in
weight. This up-thrust that the object experiences is due to the upward force
of the liquid on the object. The up-thrust therefore makes the object to
experience a loss in weight and becomes lighter when it is in the liquid. One
can therefore use one hand to lift the object out of the liquid.
Upthrust
An upthrust is an upward force that a
fluid or liquid exerts on an object that is immersed in the liquid. The unit of
upthrust is newton because upthrust is a force.
Loss
in Weight of an Object
loss in weight is the reduction in the
weight of an object when the object is partially or totally immersed in a
liquid. Loss in weight is caused by the upthrust of the liquid on the object.
Relationship
between Upthrust and Loss in Weight
Upthrust and loss in weight are
related in the sense that loss in weight of an object is determined by the
amount of upthrust that the liquid exerts on the object immersed in it. You can
also say that upthrust equals loss in weight.
This relationship is the formula that
connect the two of them together. You must also know that this relationship is
the difference in the weight of the object measured in air and when it is
immersed in a liquid. This you will see in the formula below.
you must also know that,
upthrust = loss in weight = weight or volume
of liquid displaced
Formula
for Calculating Upthrust:
Upthrust = loss in weight
= weight in air – weight in liquid
Formula
for Calculating Upthrust of Liquid on Object Immersed in the Liquid:
Upthrust
= volume of object * density of liquid * acc. due to gravity
Upthrust U = V * P * g
Note:
If the object is partially
immersed in the liquid, one-third of its volume is used in the calculation.
Therefore,
Upthrust = 1/3* volume of object * density of liquid * acc. due to
gravity
Upthrust U =1/3*V * P * g
Worked
Examples:
1.
Calculate
the upthrust of a liquid of density 0.256kg/m³ on 5.12kg mass whose density is
0.035kg/m³ completely immersed in a liquid. ( g = 10m/s sqr.)
Solution:
Data given in the question:
Mass
= 5.12kg; density of object =
0.035kg/m³;
density of liquid =
0.256kg/m³; g = 10m/s sqr.
formula:
Upthrust = volume of object * density of liquid * acc.
due to gravity
Upthrust
U = V * P * g
Note:
Volume of the object is
not given in the question. So you must first calculate the volume of the object using the formula:
Density = mass / volume
Therefore,
0.035
= 5.12 * volume
Volume
= 0.035 / 5.12; volume = 0.00684 m³
Substitution:
Upthrust U = 0.00684 * 0.256 * 10
Upthrust u = 0.0175N
Worked
Examples:
1.
An
object weighs 24.5N in air and 20.34N when immersed in a liquid of unknown
density. Calculate the upthrust of the li
2.
quid
and the loss in weight.
Solution:
Data
given in the question:
Weight
in air = 24.5N; weight in liquid =
20.34N upthrust = ?
Formula:
Upthrust
= weight in air – weight in liquid
Substitution:
i.
Upthrust
of liquid on object:
Upthrust = 24.5 – 20.34
Upthrust
= 4.16N.
ii.
Loss
in weight of object:
Loss in weight = upthrust
= weight in air – weight in liquid.
Therefore,
Loss
in weight = 4.16N
*** more worked examples would be done
latter***
Archimedes’
Principle
Aarchimedes’ principle states that
when an object is partially or totally immersed in a iquid, the object experiences
an upthrust which is equal to the weight or volume of the liquid displaced.
Verification
of Archimedes’ Principle
Aim:
to verify Archimedes principle.
Apparatus:
eureka can, beaker, object, thread and
liquid or water.
Set-up
diagram:
……….
Procedures:
….
Observation:
……………..
Conclusion:
To conclude the experiment, if the apparent loss in weight of the object, or
the up-thrust of the liquid on the object is equal to the weight of the liquid
displaced, then Archimedes’ principle is verified.
Worked
Examples on Archimedes’ Principle
1.
An
iron cube of volume 790cm³ is totally immersed in (a) water; (b) oil of density
0.75g/cm³; (c) oxygen gas of density 0.0015g/cm³. calculate the upthrust in each
case, if the weight of 1g mass is 0.01N
Solution:
Data
given in the question:
Volume of object = 790cm³;
density of oil = 0.75g/cm³; 1g = 0.01N
density of oxygen =
0.0015g/cm³.
(i)
Upthrust
of water on object:
Formula:
Upthrust,
U = volume*density* acc due to gravity
Upthrust,
U = 790 * 1 * 0.01
Upthrust,
U = 7.9 N
(ii)
Upthrust
of oil of density 0.9 g/cm3 on object:
Upthrust, U =
volume*density* acc due to gravity
Upthrust,
U = 790 * 0.9 * 0.01
Upthrust,
U = 7.11 N
(iii)
Upthrust
of nitrogen gas of density 0.02 g/cm3 on object:
Upthrust, U =
volume*density* acc due to gravity
Upthrust,
U = 790 * 0.02 * 0.01
Upthrust, U = 0.158 N
2. An empty density bottle weighs 20.0g.
it weighs 50g when filled with water and 45.2g when filled with a liquid.
Calculate the density of the liquid.
Solution:
Data
given in the question:
Mass
of empty bottle = 20.0g, mass of bottle + water = 50g,
Mass
of bottle + liquid = 45.2g.
Mass
of water =Mb+w - Mb = 50 – 20 = 30g
Mass of liquid =Mb+l - Mb = 45.3 – 20 = 25.3g
You have to first calculate the
relative density of the substance and then change it to its density.
Formula:
Relative
density = mass of substance / mass of equal volume of water
Substitution:
Relative
density = 25.3g / 30g
Relative
density = 0.8433
Now, you have to find the density of
the liquid using the formula,
Relative
density = density of substance / density of water
Substitution:
0.8433
= density of substance / 1000kg/m3
0.8433
* 1000 = density of liquid
Density
of liquid = 0.8433 * 1000
Density
of liquid = 843.3kg/m3
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