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Consider equation (4) c = Arg/l
[L.H.S] = [c] = LT-1
[R.H.S] = [ rg/l ] = (ML-3MT-2)/L =
M2L-4 T-2
Equation (4) is dimensionally inconsistent and therefore
wrong.
Note: Since equation (3) is the only one which is
dimensionally consistent, it is the correct one.
(ii) Using equation (3) c2 = Ag/rl
c2 = ( Ag/r)1/l of the form y = mx
A graph of c2 against 1/l should be linear
with a slope s = Ag/r
| c (ms-1) |
0.67 |
0.45 |
0.36 |
0.27 |
| c2(m2s-2 ) |
0.45 |
0.20 |
0.13 |
0.07 |
| l x 10-3(m) |
1.0 |
2.2 |
3.5 |
6.1 |
| 1/l x102(m-1 ) |
10.0 |
4.55 |
2.86 |
1.64 |
(The student should plot a graph of c2 against
1/l)
The graph is linear, which confirms the choice of equation
(3).
Slope s = 44 x 10-5 m3s-2
Using A = sr/g = (44 x 10-5 x 103 )/
(7.2 x 10-2 ) = 6.1
N.B: It is possible for more than one equation to
be dimensionally consistent. Graphical methods can then be
used to identify the correct equation out of all the possible
equations.(see example 2 below.)
Example 2:
In an attempt to describe the variation of pressure P
with velocity v in the streamline flow of a non- viscous
incompressible fluid, along a horizontal pipe, a student formulated
the following equations:
1. P + Ag rv = X
2. P + Brv2 = Y
3. P + cggv-2 = Z
In which A, B and C are dimensionless constants. X, Y and
Z are constants with the dimensions of pressure, g
is the acceleration due to gravity, r is the density
of the liquid, g is the coefficient of surface tension
of the liquid.
(i) Which of the equations are dimensionally consistent?
(ii) The following results were obtained in an experiment
in which P and v were measured for water flowing
along a pipe of varying area of cross-section.
| P x 103 Nm-2 |
2.0 |
1.5 |
1.2 |
0.7 |
0.3 |
| v (ms-1 ) |
1.0 |
1.4 |
1.6 |
1.9 |
2.1 |
Use these results to distinguish which of the above equations
is correct and find the value of the constants in the equation.
(Density of water = 1.0 x 103 kg m-3,
coefficient of surface tension of water = 7.4 x 10-2 Nm-1
)
SOLUTION
[P] = ML-1 T-2 = [X] = [Y] = [Z]
[g] = LT-2, [r] = ML-3, [v] = LT-1
Consider the dimensional consistency of each equation.
1. P + Agrv = X
P - X = - Agrv
[L.H.S] = [(P - X)] = ML-1T-2
[R.H.S] = LT-2 x ML-3 x LT-1
= ML-1 T-3
\ [L.H.S] ¹ [R.H.S]. Hence equation (1) is dimensionally
inconsistent and is therefore wrong.
2. P + Brv2 = Y
P - Y = Brv2
[L.H.S] = [(P - Y)] = ML-1 T-2
[R.H.S] = [rv2 ] = ML-3 x L2
T-2 = ML-1 T-2
[L.H.S] = [L.H.S]. \ equation (2) is dimensionally consistent
and is possibly correct.
3. P + Cgv-2 = Z
P - Z = - Cggv-2
[L.H.S] = [P - Z] = ML-1 T-2
[R.H.S] = MT-2 x LT-2 x L-2
T2 = ML-1 T-2
\ equation (3) is dimensionally consistent and is possibly
correct.
Either equation (2) or equation (3) is correct since both
are dimensionally consistent. The correct equation can be
identified by a graphical method.
Suppose equation (2) is the correct one, then
P + Brv2 = Y
P = - Brv2 + Y is of the form y = mx + c where
y º P, m º - Br and c º Y
A graph of P against v2 should be
a straight line with a P - intercept = Y and a gradient =
-Br
Suppose equation (3) is the correct one, then
P + Cggv-2 = Z
P = - (Cgg) 1/v2 + Z of the form y = mx + c
A graph of P against v-2
should be a straight line of gradient = - Cgg and with a P
- intercept = Z.
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