Dimensional Analysis: Finding Relation Between v, p and p

[look416]

Homework Statement

The velocity,v of the sound in a gas depends on the density, p and the pressure, p of the gas. By using dimensional analysis, find a possible relation between v, p and p.

Homework Equations

[v] = LT^-1
[p] = ML^-3
[p] = ML^-1T^-2

The Attempt at a Solution

well i have no idea how to do it, so far i can only express the variables in dimension ways

 

[RoyalCat]

Here, have a ρ to express the density.

The whole concept of dimensional analysis is based on the fact that you can only equate two quantities if they have the same dimensions and units.

You are told that the velocity of sound in a gas, $$v$$ depends on two factors, that is to say, that it is a function of these two factors.

$$v=f(\rho , p)$$

Naively, we say that it is a product of these two quantities, raised to some powers $$\alpha, \beta$$:

$$v=\rho^{\alpha}\cdot p^{\beta}$$

Now use dimensional analysis to find $$\alpha, \beta$$

Do this by constraining the system so that the dimensions on the RHS are the same as the dimensions on the LHS.

 

[tomcue]

, such as v = LT-1, p = ML-3, and p = ML-1T-2. However, by using dimensional analysis, we can manipulate these equations to find a relationship between v, p, and p. This can be done by setting up a dimensional equation with the given units and using algebraic manipulation to solve for the unknown relationship. In this case, we can write:

v = k p^a p^b

Where k is a constant and a and b are exponents to be determined. By equating the dimensions on both sides, we get:

LT^-1 = (ML^-3)^a (ML^-1T^-2)^b

Simplifying, we get:

LT^-1 = M^(a+b) L^(-3a-b) T^(-2b)

Equating the powers of M, L, and T on both sides, we get the following system of equations:

a + b = 0
-3a - b = -1
-2b = -1

Solving for a and b, we get:

a = 1/2
b = -1/2

Therefore, the relationship between v, p, and p is:

v = kp^(1/2)p^(-1/2)

Or, simplifying:

v = kp^0

v = k

This means that the velocity of sound in a gas is directly proportional to a constant k, independent of the density and pressure of the gas. In other words, the velocity of sound in a gas is constant as long as the gas remains in the same state. This relationship can also be verified experimentally by measuring the velocity of sound in different gases at the same pressure and density.