If p is a pressure, V a velocity, and ρa fluid density, what are the dimensions (in the M L T sy

If p is a pressure, V a velocity, and ρa fluid density, what...

Key Concepts

Density

Density is a measure of mass per unit volume. With mass being represented by M and volume by L³, the dimensions of density are M L?³. This basic concept is important in understanding properties of materials and fluids, and in deriving other physical quantities in fluid mechanics and material science.

Velocity

Velocity represents the rate of change of displacement with respect to time. Its dimensions are expressed as length per time (L T?¹), indicating how quickly an object or fluid element moves through space. This fundamental concept is essential in analyzing movement and dynamic processes in various physical systems.

Pressure

Pressure is a measure of force distributed over an area. Since force is defined as mass times acceleration (where acceleration is length per time squared), the dimensions of pressure become mass divided by length and time squared, represented as M L?¹ T?². This concept is crucial in many fields including fluid dynamics and thermodynamics.

Dimensional Analysis

Dimensional analysis is a fundamental method in physics used to deduce the relationships between physical quantities by comparing their dimensions. By expressing quantities in terms of the basic dimensions of mass (M), length (L), and time (T), this technique ensures that equations are dimensionally consistent and helps in the derivation of new relationships between physical variables.

Transcript

00:02 So in this question, what we are required to do is that we are required to find out the dimensions, right? and for that, we have been given three cases.

00:14 The first case is that we have a ratio of pressure to density.

00:19 In the second case, we have a product that is of pressure, volume, and density.

00:27 And in the third part, we need to find out the dimension.

00:32 And that is again a ratio right so that is p divided by the product of density times volume square right so for that what we are going to do that the first tip is to analyze the dimensions for the quantities right so we have a dimensional dimensional analysis right so let's take a look at that.

01:07 Now in the first column we have the physical quantities right and in the second column we have the dimensions in the mlt system right.

01:31 So this is our first row.

01:37 Right.

01:37 So in this first row, the first row, the first, this is.

01:42 This is.

01:42 The first row right so in the second row we have the first quantity that is pressure right so the unit of pressure is newton per meter square right so now that we have the unit of the pressure as well so we can easily determine the dimension so the dimension for um pressure it's mass per length per time square right so this is dimension for pressure now next moving on we have the density right so density is simply row right and the unit of density is kilogram per meter cube right and the dimension of density is m per length cube right then moving forward we have volume right so the unit of volume is meter cube right and the dimension of volume is the length cube right so since we had the units as well oh right so it made us um it made easy to find out the dimension of all of these three quantities.

03:14 So now that we have the dimensions, it's easy for us to find out the dimension for each case.

03:20 So this table is very important, right? so let's start with the first case.

03:28 So in this case, that is the first case, we have a ratio that is being divided by row.

03:36 Right...

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