24 july 2014 K.N. Chattopadhyay

2321-7871
Weekly Science Research Journal
Vol-2, Issue-2, 24th July 2014
Original Article
An Improvised Method For The Determination Of
Density Of Solids
K.N. Chattopadhyay
ABSTRACT
K.N. Chattopadhyay
Density of a substance is the mass of the substance per unit volume. The conventional
method for the determination of the density of a substance is to measure its mass with
the help of a physical balance and its volume by measuring cylinder. But there are
many alternative methods also where density of solids and liquids can be determined by
without measuring its mass and volume directly. In this paper another improvised
method for the determination of solids is described. In this method also the density of
solids can be determined by without measuring the mass and volume; it is determined
only with the help of a meter scale, a test tube and a liquid of known density. The
advantage of this method is that it is based only on the principle of moment of force and
can be used not only for the solids of appreciable sizes but also for solids of small sizes
or are of powder form.
Keywords:
Determination Of Density , physical balance , measuring cylinder ,Improvised
Method
1.Introduction:
Density of a substance is a physical quantity which is defined as mass
of unit volume of the substance. There are many methods for the
determination of density of substances. To find out the density we need a
physical balance for the measurement of mass and a measuring cylinder for
volume. But there are other many methods where the density of substances is
determined without measuring the mass and volume directly. For example,
Mumba and Tsige [1] have described a method where the density of a solid can
be measured with the help of a meter scale only. In this method, they have used
two principles: Archimedes principle and the principle of moment of force.
Chattopadhyay has also developed the alternative methods [2,3] for the
determination of density of liquid without measuring its mass and volume. The
method developed by Mumba and Tsige [1] is based on Archimedes principle
and is thus not suitable for the determination of density of solids which are of
very small sizes (e.g., sand) and which are in the form of powder, because the
volume of solid of small size is very less and so when it is immersed completely
in water, the up thrust acting on it is so small that it can not bring an
appreciable change in the meter scale reading. In this paper another method
for the determination of density of any type of solids, without measuring the
mass and volume directly, is described where only the principle of moment of
force is used and Archimedes principle is not needed and thus suitable for the
solids of small sizes or in powder form.
Experimental procedure
The experimental set up is shown in figure-1. It is a very simple
arrangement. Figure-1 shows that a meter scale is hung by a string from a rigid
support at center point A. A solid substance of any shape and of unknown
mass is hung from the left side of the meter scale with a string at point X. An
empty test tube of unknown mass is hung from a string at the right hand side of
Page No-1
From
Department of Education, The
University of Burdwan, Golapbag,
Burdwan, West Bengal , India.
Article Is Published On July 2014 Issue
& Available At www.weeklyscience.org
DOI : 10.9780/ 2321-7871/1202013/53
2321-7871
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Vol-2, Issue-2, 24 July 2014
An Improvised Method For The Determination Of Density Of Solids
the scale and the position of test tube is so chosen that the system is balanced. Let this
position of the empty test tube be Y. If the masses of the solid substance and the empty test
tube are M and m respectively and the distances of points X and Y from point A are L and l
respectively, then from the principle of moments of forces we can write:
MgL = mgl
(1)
,
Figure 1. Balancing of empty test tube
Now water (whose density is known) is poured in the test tube up to mark O. The
system becomes unbalanced. To balance the system, the water-filled test tube is moved to a
point Y/ on the right side of the meter scale (figure 2) but the position of the solid substance
remain same. If the distance of point Y/ from point A is l’ then again from the principle of
moment of forces we can write:
MgL = (m+Ml)gl’
(2)
where Ml is the mass of water in the test tube.
Figure 2. Balancing of water-filled test tube
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An Improvised Method For The Determination Of Density Of Solids
The next step is to remove the water from the test tube and to place the solid
substance (bigger size or smaller size) whose density is to be determined in the test tube in
such a way so that the test tube is partially filled. The system is again unbalanced and to
balance it the test tube is moved to the position Y// in the meter scale ( keeping the solid
substance in the left side in its original position X) (figure 3). If this balancing length (AY//) is
l” then from the principle of moment of forces we have:
MgL = (m+Ms)g l”
(3)
where Ms is the mass of the solid substance .
Figure 3. Balancing of test tube partially filled with the solid substance
Now keeping the solid substance inside the test tube, the remaining portion of the
tube is filled by water (up to the mark O) very carefully and the tube is again balanced by
moving it to another position Y/// (figure 4). If the balancing length be l”’ in this case, then
again from the principle of moment of force we can write
MgL = (m+Ms +M2 )g l”’
(4)
Where M2 is the mass of the water in the remaining portion of the tube.
Now from equations (1) and (2) we have
Mlgl’ = mg ( l- l’ )
(5)
If V be the volume of the space inside the tube up to mark O, and dl is the density of water in
the room temperature, then M1 = Vdl. So, equation (5) can be written as
Vdlgl’ = mg ( l- l’ )
(6)
Similarly, from equations (1) and (3) we have,
Msg l” = mg ( l- l”)
(7)
If Vs and ds be the volume and density of the solid substance respectively,
then Ms = Vs ds. So equation (7) can be re written as
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An Improvised Method For The Determination Of Density Of Solids
Figure 4. Balancing of test tube filled with the solid substance and water
Vsds g l” = mg ( l- l”)
(8)
From equations (1) and (4) we have
( Ms +M2) g l”’ = mg (l- l”’ )
(9)
Now M2 = (V-Vs) d1 . So, equation (9) takes the form
[Vsds + (V-Vs) d1] g l”’ = mg (l- l”’ )
(10)
From the above equation we can write
Vsd1 = Vsds + Vd1 – m(l- l”’ )/ l”’
= m ( l- l”)/ l” + m ( l- l” )/ l” - m(l- l”’ )/ l”’
[ As from eq(8) Vsds = m ( l- l”)/ l” and from eq (6) Vd1 = m ( l- l’ )/ l’ ]
or,
Vsd1 = m ( l/ l’ + l/ l” - l/ l”’ - 1)
(11)
And from equation(8)
Vsds = m ( l- l”)/ l” = m (l/ l” - 1)
(12)
From equations (11) and (12) we can write
ds/d1 =
(l/ l” - 1)/ ( l/ l’ + l/ l” - l/ l”’ - 1)
(13)
As d1 is known, from the above equation ds can be calculated by knowing only the three
balancing lengths l’, l” and l”’ .
Discussion
From equation (13) it is clear to us that the density of solid substance can be
determined without measuring its mass and volume. The advantage of this method is that
here the test tube acts as the container and this method is not based on Archimedes
principle and thus the size of the solid does not matter. Using this method we have
measured the densities of common solids like iron, copper, marble etc and sand. The
results agree well with the already known values which proves the accuracy of this method.
But for those solids which are lighter than water, this method is not suitable. In this
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situation one can use the method developed by Chattopadhyay [4]. If a solid shows
chemical reaction with water or dissolves in water then water may be replaced by another
suitable liquid of known density.
This simple and accurate method of finding the density of solid is useful in schools
where there is no physical balance and measuring cylinder. Not only that this method can
also be treated as improvised method and can be used for the students of well developed
schools as a project which leads to joyful learning and helps in enhancing their creativity.
References:
1.Mumba F and Tsige M 2007 Phys. Educ. 42 , 293-5
2.Chattopadhyay K.N. 2008 Phys. Educ. 43, 203-5.
3.Chattopadhyay K.N. 2012 Lat. Am. J. Phys. Educ. Vol. 6, Suppl. I, 407-9
4.Chattopadhyay K.N. 2009 Phys. Educ.44,107(2009), erratum 44, 208.
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