How to apply momentum equation to wedge with angle θ

[Saladsamurai]

Homework Statement

Homework Equations

$$\sum(\dot{m}\vec{v})_{exit}-\sum(\dot{m}\vec{v})_{inlet}=\sum F_{ext}$$


I only know that this is cons of momentum because my prof told us. I am having a hard time visualizing how to incorporate THETA into the above equation since it is not a nice right triangle.

I am thinking that since the exit velocities are equal, than the inlet flow must be being split about the wedges horizontal axis of symmetry.

Thus, I think I can divide the wedge into an "upper" and "lower" right triangle whose angle wrt to the horizontal is $\frac{\theta}{2}$.

Then I can resolve the momentum eq into Cartesian coordinates.

Sound good?

 

[Doc Al]

Sounds lovely!

 

[Saladsamurai]

Nice! I guess that what is confusing me now, is why they gave me
force/unit length into page

But I am just going to start plugging in and see what happens.

 

[Doc Al]

I guess that what is confusing me now, is why they gave me
force/unit length into page

Consider it to be an arbitrarily wide sheet of water. Use momentum per unit length as well.

 

[Saladsamurai]

Consider it to be an arbitrarily wide sheet of water. Use momentum per unit length as well.

Okay, so the m^-1 just drops out anyway, and this problem reduces to regular cons of mom

 

[Doc Al]

Yep.

 

[Saladsamurai]

Thanks! 83.4 degrees sounds reasonable I think

 

[Doc Al]

83.4 degrees sounds reasonable I think

Show how you got that answer.

 

[Saladsamurai]

 

[Doc Al]

Rethink your result for m, the mass flow rate. You want it to be mass flow per unit width (or depth).

 

[Saladsamurai]

I am not sure that I follow. Errr... Okay. So that inlet with the 4 cm dimension is NOT a PIPE...right?

It is a "sheet" with area 4cm*WIDTH. Thus my, mass flow rate should be, with h=4cm:

$$\dot{m}=\rho V (h*W)\Rightarrow \frac{\dot{m}}{W}=\rho Vh$$

Am I with you now?

 

[Doc Al]

Now you're cooking.

 

[Saladsamurai]

Typical 'not-paying-attention' mistake. Perhaps I should turn off Band of Brothers while I do my studies?

Thanks Doc!