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Example:
A student in S.5 recalls the following equations of motion.
1. v = u + at2
3. v2 = u2
+ 2as
2. s = ut + ½ at2
4. s = ut + 2at2
where v is final velocity, u is initial velocity,
a is acceleration, s is distance
and t is time.
Check the dimensional consistency of the above equations
and comment on your answers.
SOLUTION
Consider eq.(1)
v = u + at2
[R.H.S] = [u] + [at2 ] = LT-1 + (LT-2
x T2 ) = LT-1 + L
[L.H.S] = [v] = LT-1
[R.H.S] ¹ [L.H.S] \ equation (1) is obviously wrong.
Consider equation (2)
s = ut + ½ at2
[L.H.S] = [s] = L
[R.H.S] = [ut ] + [at2 ] = (LT-1 x
T) + (LT-2 x T2 ) = L + L = L
[R.H.S] = [L.H.S] \ equation (2) is possibly correct.
Consider equation (3)
v2 = u2 + 2as
[L.H.S] = [v2 ] = (LT-1 )2
= L2 T-2
[R.H.S] = [u2 ] + [as] = L2 T-2
+ L2 T-2 = L2 T-2
[R.H.S] = [L.H.S] \ equation (3) is possibly correct.
Consider equation (4)
s = ut + 2at2
[L.H.S] = [s] = L
[R.H.S] = [ut] + [at2 ] = LT-1 T +
LT-2 T2 = L
[R.H.S] = [L.H.S] \ equation (4) is possibly correct.
The student should note:
1. Equation (2) and equation (3) are dimensionally consistent
and correct.
2. Equation (4) is dimensionally consistent but wrong. Dimensional
consistency does not prove the correctness of an equation.
3. The check for the consistency of dimensions does not provide
any information about the correctness of numerical factors
like the ½ in equation (2) or in equation (4).
Checking for
correctness of an equation:
After eliminating the obviously wrong equations,the correct
equation can then be obtained from the remaining equation
by a graphical method.
Example 1:
The velocity of propagation, c, of ripples on the surface
of a liquid is given by one of the following equations.
1. c2 = Arl/g
2. c = Arlg2
3. c2 = Ag/rl
4. c = Arg/l
where A is a dimensionless constant, gis the
surface tension of the liquid, r is its density and
l is the wavelength of the ripples.
(i) Use the method of dimensions to determine which equation
is correct.
(ii) By a graphical method, use the following figures for
water to confirm your choice, and to determine the value of
A
.
| c (ms-1 ) |
0.67 |
0.45 |
0.36 |
0.27 |
| l x 10-3 (m) |
1.0 |
2.2 |
3.5 |
6.1 |
(coefficient of surface tension of water = 7.2 x 10-2
Nm-1 and density of water = 103 kg/m3
)
SOLUTION
[c] = LT-1 , [r] = ML-3 ,[l] = L, g
= force per unit length.
[g] = [force]/[length] = MLT-2/L = MT-2
Consider equation (1) c2
= Arl/g
[L.H.S] = [c2] = (LT-1 )2
= L2 T-2
[R.H.S] = [ rl/g] = ML-3L = L-2 T2
[L.H.S] ¹ [R.H.S] \ equation (1) is wrong.
Consider equation (2)
c = Arlg2
[L.H.S] = [c] = LT-1
[R.H.S] = [rlg2] = ML-3LM2T-4
= M3L-2T-4
[L.H.S] ¹ [R.H.S].
\ equation (2) is wrong.
Consider equation (3) c2
= Ag/rl
[L.H.S] = [c2] = L2 T-2
[R.H.S] = [ g/rl] = MT-2/ML-3L = L2
T-2
Equation (3) is dimensionally consistent.
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