Dimensional analysis problem

[Michael_Light]

Homework Statement

The equation P+av²=b is dimensionally correct. P and v represent pressure and velocity respectively. The unit of ''a'' is similar to...?

The answer provided is density...

Homework Equations

The Attempt at a Solution

In the question... it doesn't state that what is ''b'', so should i assume that it is a constant? or there's a proper solution? Please enlighten me...

 

[Delphi51]

There is no telling if b is constant or not.
All you can say is that it has the units of pressure, so it is "similar" to pressure.

 

[Michael_Light]

There is no telling if b is constant or not.
All you can say is that it has the units of pressure, so it is "similar" to pressure.

How do you know that ''b'' has a unit of pressure in the first place? Thanks...

 

[Delphi51]

The formula P+av²=b has three terms. All of them must have the same units or dimensions because you can't add kg to meters. P is in units of pressure, so av² and b must also be in units of pressure.

 

[rwisz]

It is possible to solve this problem without knowing the value of b, as long as we understand the concept of dimensional analysis. Dimensional analysis is a method used to check the consistency of equations by comparing the units of each term. In this case, we can compare the units of P and av^2 on both sides of the equation.

On the left side, P represents pressure, which has the unit of force per unit area (N/m^2 or Pa). On the right side, av^2 represents the product of acceleration (m/s^2) and velocity squared (m^2/s^2), which also has the unit of force per unit area. This means that the two sides of the equation are consistent in terms of units.

Now, let's look at the unit of 'a'. Since av^2 represents force per unit area, we can rearrange the equation to get a = (b-P)/v^2. This means that the unit of 'a' must be force divided by velocity squared (N/m^2 * s^2). This unit is not similar to density, which is mass per unit volume (kg/m^3). Therefore, the answer provided is not correct.

In conclusion, the unit of 'a' in this equation is N/m^2 * s^2 and it is not similar to density.