The volume of liquid flowing per second is called the volume flow rate "Q" and has the dimensions of [L]^3/[T]. The flow rate of a liquid through a hypodermic needle during an injection can be estimated with the following equation.
Q = pi * R^n (P2 - P1) /
8ηL
The length and radius of the needle are L and R, respectively, both of which have the dimension [L]. The pressures at opposite ends of the needle are P2 and P1 both of which have the dimensions of [M]/{[L][T]}. The symbol pi stands for pi and like the number 8 and the exponent n, has no dimensions. using dimensional analysis, determine the value of n in the expression for Q.
n=4
To determine the value of n in the expression for Q using dimensional analysis, we need to equate the dimensions on both sides of the equation.
Starting with the expression for Q:
Q = π * R^n * (P2 - P1) / (8ηL)
Let's break down the dimensions of each term:
- Q has the dimensions of [L]^3/[T]
- π has no dimensions
- R^n has the dimensions of [L]^n
- (P2 - P1) has the dimensions of [M]/{[L][T]}
- 8 has no dimensions
- η has the dimensions of [M]/([L][T])
- L has the dimensions of [L]
Now, we can set up the equation for the dimensions:
[L]^3/[T] = [L]^n * [M]/{[L][T]} / ([L][T])
To simplify, let's divide both sides of the equation by [L][T]:
[L]^2/[T]^2 = [L]^n * [M]/{[L][T]} / [T]
Next, let's rearrange the equation by shifting [L]^n to the left side:
[L]^2/[L]^n = [M]/{[L][T]} / [T]
Now we can simplify the left side by dividing [L]^2 by [L]^n:
[L]^(2-n) = [M]/{[L][T]} / [T]
To equate the dimensions on both sides, the terms in the numerator and denominator must cancel each other out. Since [M]/{[L][T]} / [T] has the dimension of [M]/{[L]^2[T]^2}, we can rewrite the equation as:
[L]^(2-n) = [M]/{[L]^2[T]^2}
Now, because dimensions on each side of the equation need to be equal, the exponents of [L] must be the same. Therefore, we can equate the exponents:
2 - n = 2
Now, solve for n:
n = 2 - 2
n = 0
Thus, the value of n in the expression for Q is 0.
To determine the value of the exponent "n" in the expression for Q using dimensional analysis, we analyze the dimensions of each term in the equation.
1. Q: Volume flow rate = [L]^3/[T]
2. pi (π): Dimensionless
3. R: Radius of the needle = [L]
4. P2 - P1: Pressure difference = [M]/([L][T])
5. η: Viscosity = [M]/([L][T])
6. L: Length of the needle = [L]
Now, let's evaluate the dimensions of each term in the equation:
Q = pi * R^n * (P2 - P1) / (8ηL)
Dimension of the left-hand side = [L]^3/[T]
Dimension of the right-hand side:
pi * R^n * (P2 - P1) / (8ηL)
= Dimensionless * [L]^n * [M]/([L][T]) / ([M]/([L][T])) * [L]
Simplifying the above expression:
[L][T] * [L]^n / ([L][T]) * [L]
= [L]^n
Since the left-hand side must have the same dimension as the right-hand side, we equate the dimensions:
[L]^3/[T] = [L]^n
Now, we can conclude that:
n = 3
Therefore, the value of the exponent "n" in the expression for Q is 3.