Duct Pressure Question: Troubleshooting Complicated Units and Conversions

[cwill53]

Homework Statement

A vacuum gage at the intake duct to a fan gives a reading of 6 in of water. The surrounding atmospheric pressure is $14.5 lbf/in^2$ Determine the absolute pressure inside the duct, in $lbf/in^2.$ The density of water is $62.39lb/ft^3$, and acceleration of gravity is $32.0 ft/s^2$.

Relevant Equations

$$p=p_{atm}+ \rho gL$$

These units complicate everything and I simply cannot get them to check out for the life of me. After making some conversions I got to this point:

 

[haruspex]

Ask yourself what a vacuum gauge measures.
Wrt units, how do you convert lbm to lbf? (You could have saved yourself some arithmetic.)
How do you convert ft² to in²?

 

[cwill53]

Ask yourself what a vacuum gauge measures.
Wrt units, how do you convert lbm to lbf? (You could have saved yourself some arithmetic.)
How do you convert ft² to in²?

I ended up solving the problem by realizing 1 lbm= 1 lbf on Earth essentially and the density can be expressed in terms of $lbf/ft^3$.
Thanks a lot.

 

[haruspex]

I ended up solving the problem by realizing 1 lbm= 1 lbf on Earth essentially and the density can be expressed in terms of $lbf/ft^3$.
Thanks a lot.

What answer did you get? You did not respond to my remark about what a vacuum gauge measures.

 

[Chestermiller]

The correct starting equation, in proper English units should be $$p=p_{atm}-\frac{\rho gL}{g_c}$$where $$g_c=32\ \frac{lb_m\ ft}{lb_f\ sec^2}$$So, $$p=p_{atm}-\frac{(62.4)(32)(0.5)}{32}\frac{1\ ft^2}{144\ in^2}=14.5-0.2=14.3\ psi$$

 

[cwill53]

What answer did you get? You did not respond to my remark about what a vacuum gauge measures.

I got the answer of $14.2834lbf/in^2$

A vacuum gauge measures pressures that are lower than atmospheric pressure.