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**The Speed of Sound V in a Gas might plausibly depend on the Pressure P, the Density Ρ, and the Volume V of the Gas. Use Dimensional Analysis To Determine The Exponents X, Y, And Z in the Formula **

**V = Cp**^{x}**ρ**^{y}**v**^{z}

**where C is a dimensionless constant. Incidentally, The mks units of pressure are kilograms per meter per second squared. Solution:-**Equating the dimensions of both sides of the above equation, we obtain

A comparison of the exponents of [L], [M], and [T] on either side of the above expression yields,

1 = -x – 3y +3z

0 = x+y

-1 = -2x

The third equation immediately gives x = ½ ; the second equation then yields y = – ½ ; finally, the first equation gives z = 0. Hence,

**Problem 2:-****Milk is flowing through a full pipe whose diameter is known to be 1.8 cm. The only measure available is a tank calibrated in cubic feet, and it is found that it takes 1 h to fill 12.4 ft ^{3}.**

velocity is [L]/[t] and the units in the SI system for velocity are therefore m s

v = L/t where v is the velocity.

Now V = AL where V is the volume of a length of pipe L of cross-sectional area A

i.e. L = V/A.

Therefore v = V/At

Checking this dimensionally

[L][t]

which is correct.

Since the required velocity is in m s

From the volume measurement

V/t = 12.4ft

We know that,

1 ft

1 = (0.0283 m

1 h = 60 x 60 s

So, (1 h/3600 s) = 1

Therefore V/t = 12.4 ft

= 9.75 x 10

Also the area of the pipe A = πD

= π(0.018)

= 2.54 x 10

v = V/t x 1/A

= 9.75 x 10

= 0.38 m s

Let F ∝ v

So, F = Kv

where ‘K’ is a dimensional constant.

Dimensional formula of F = [M

Dimensional formula of v = [L

Dimensional formula of r = [L

Dimensional formula of η = [M

Substitute for the dimensional formulae in equation (1),

[M

[M

In accordance to principle of homogeneity, the dimensions of the two sides of relation (2) should be same.

So, c = 1 …... (3)

a+b-c = 1 …... (4)

-a – c = -2 …... (5)

Putting c = 1 in (5), we get a = 1

Putting a = 1 and c = 1 in (4), we get b = 1

Substituting for a, b and c in (1), F = kηrv, which is required relation.

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