Volume of Liquid: Proving $\overrightarrow{v}\cdot\overrightarrow{n}$

In summary, the conversation discusses the relationship between a liquid flowing through a flat surface with uniform velocity and the volume of liquid passing through a unit surface in a unit of time. The use of the dot product is suggested as a way to show this relationship, with reference to a Wikipedia article on volumetric flow rate.
  • #1
mathmari
Gold Member
MHB
5,049
7
Hey! :eek:

A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$.

Let $\overrightarrow{n}$ an unit vector perpendicular to the plane.

Show that $\overrightarrow{v} \cdot \overrightarrow{n}$ is the volume of the liquid that passes through the unit surface of the plane in the unit of time.

Could you give me some hints how we could show this?? (Wondering)
 
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  • #2
I found the following:

Volumetric flow rate - Wikipedia, the free encyclopedia

To show that $\overrightarrow{v} \cdot \overrightarrow{n}$ is the volume of the liquid that passes through the unit surface of the plane in the unit of time, do we use the justification at the part "The reason for the dot product is as follows" of wikipedia?? (Wondering)
 
  • #3
Yep. (Nod)
 

FAQ: Volume of Liquid: Proving $\overrightarrow{v}\cdot\overrightarrow{n}$

What is the volume of liquid and why is it important?

The volume of liquid is the amount of space that a liquid occupies. It is important because it is a key factor in determining the amount of a substance present and can help calculate other properties such as density and mass.

What is the formula for calculating volume of liquid?

The formula for calculating volume of liquid is V = A x d, where V is the volume, A is the surface area, and d is the depth or height of the liquid.

What is the significance of proving $\overrightarrow{v}\cdot\overrightarrow{n}$ when calculating volume of liquid?

Proving $\overrightarrow{v}\cdot\overrightarrow{n}$ is important because it allows us to determine the direction and magnitude of the flow of liquid, which is crucial in accurately calculating the volume.

How do you measure the volume of irregular shaped liquids?

To measure the volume of irregular shaped liquids, you can use a graduated cylinder or other measuring device. You can also use the displacement method, where you measure the volume of liquid before and after submerging an object in it, and then calculate the difference.

How is the volume of liquid affected by temperature and pressure?

The volume of liquid is affected by temperature and pressure because they can cause the liquid to expand or contract. As the temperature increases, the volume of the liquid will also increase. Similarly, an increase in pressure will decrease the volume of the liquid. These factors must be taken into account when calculating the volume of a liquid.

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