Im having severe trouble understanding the concept of Dimensions. And with 300 other students in my physics class the Professor doens't have alot of time for individual students....
The volume of a liquid flowing per second is called the volume flow rate Q and has the dimensions of [L]^3/[T]. The flow rate of the liquid through a hypodermic needle during an injection can be estimated with the following question:
Q= (pi)R^n(P2-P1)/ 8nL
The length and radius of the needle are L and R, respectively, both of which have a dimension [L]. The pressures at opposite ends of the needle are P2 and P1 , both have the dimensions of [M]/{[L][T]^2}. The symbol n (On the bottom of the fraction) has the dimensions [M]/{[L][T]}. Pi, the number 8 and the exponent n (On the top) have no dimensions. Using dimensional analysis, determine the value of n in the expression for Q.
I have honestly spent an hour on this question with lettle or no understanding of what it means or how to go about solving it. Can someone please help?
Thanks.
somehow I see an error. Mass is in the pressure unit, but nowhere else. It wont cancel if that is so.
Q= (pi)R^n(P2-P1)/ 8nL
Just substitute the dimentsions for all the symbols:
Q = L^3 T^(-1)
R = L
P2 - P1 = M L^(-1)T^(-2)
n = M L^(-1)T^(-1)
L^3 T^(-1) =
L^n M L^(-1)T^(-2)M^(-1) L T =
L^n T^(-1) ---->
n = 3
Sure! I can help you understand the concept of dimensions and how to solve this problem using dimensional analysis.
In physics, dimensions are used to describe the physical quantities and units. They represent the basic nature and type of a physical quantity. For example, length is a one-dimensional quantity, time is a one-dimensional quantity, and volume is a three-dimensional quantity.
In dimensional analysis, we analyze the dimensions of different quantities in an equation to determine the relationship between them. By comparing the dimensions on both sides of an equation, we can find the unknown dimensions or values.
Let's analyze the given equation for the volume flow rate Q:
Q = (pi)R^n(P2 - P1) / 8nL
First, let's analyze the dimensions on each side of the equation:
On the left side of the equation, the volume flow rate Q has the dimensions of [L]^3/[T].
On the right side of the equation, we have several terms:
- The term (pi) has no dimensions, so we don't need to consider it in our analysis.
- The term R^n has the dimensions [L]^n.
- The term (P2 - P1) has the dimensions [M]/([L][T]^2).
- The term 8 has no dimensions.
- The term n has the dimensions [M]/([L][T]).
- The term L has the dimensions [L].
Now, let's combine the dimensions on the right side of the equation:
R^n * [M]/([L][T]^2) * [M]/([L][T]) * [L]
= [L]^n * [M]^2 / ([L]^2 [T]^3) * [L]
= [L]^(n+1) * [M]^2 / [L]^3 [T]^3
Now, equating the dimensions on both sides of the equation, we have:
[L]^3/[T] = [L]^(n+1) * [M]^2 / [L]^3 [T]^3
To make both sides of the equation have the same dimensions, we need to equate the exponents of the dimensions [L], [M], and [T].
From the equation, we can see that:
1 = (n+1) - 3
Simplifying the equation, we get:
n + 1 - 3 = 1
n - 2 = 1
n = 3
Therefore, the value of n in the expression for Q is 3.
I hope this explanation helps you understand how to solve the problem using dimensional analysis. If you have any further questions, feel free to ask!