prove that v²=f/a*s with dimensional formula
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Answer:
The dimension of any physical quantity expresses its dependence on the base quantities as a product of symbols (or powers of symbols) representing the base quantities. (Figure) lists the base quantities and the symbols used for their dimension. For example, a measurement of length is said to have dimension L or L1, a measurement of mass has dimension M or M1, and a measurement of time has dimension T or T1. Like units, dimensions obey the rules of algebra. Thus, area is the product of two lengths and so has dimension L2, or length squared. Similarly, volume is the product of three lengths and has dimension L3, or length cubed. Speed has dimension length over time, L/T or LT–1. Volumetric mass density has dimension M/L3 or ML–3, or mass over length cubed. In general, the dimension of any physical quantity can be written as
L
a
M
b
T
c
I
d
Θ
e
N
f
J
g
for some powers
a
,
b
,
c
,
d
,
e
,
f
,
and g. We can write the dimensions of a length in this form with
a
=
1
and the remaining six powers all set equal to zero:
L
1
=
L
1
M
0
T
0
I
0
Θ
0
N
0
J
0
.
Any quantity with a dimension that can be written so that all seven powers are zero (that is, its dimension is
L
0
M
0
T
0
I
0
Θ
0
N
0
J
0
) is called dimensionless (or sometimes “of dimension 1,” because anything raised to the zero power is one). Physicists often call dimensionless quantities pure numbers.