Level sensitivity and what it equals in degrees

[Davo J]

Hi,
I posted this yesterday answering a question from a member, but the thread has been deleted because of comments.
After taking the time to work it all out and getting a PM, I thought I would post this up on it's own as I am sure others have wondered about it.

I have a few digital levels and angle gauges and over time have wondered how they compare to normal and precision machinist levels.
The standard type Stabila builders levels work out to have an accuracy of 0.75mm /per meter or 0.043 degrees

A machinist level like a Starrett No 98 is 0.005 in/foot or 0.42mm/meter per division which works out to be around  0.0239 degrees.
The level divisions can be split by eye into 5 pretty easy which then works out to be around 0.00478 degrees or better.

The more sensitive levels like the Starrett No 199 or the cheaper alternatives from China are 0.0005 in/foot or 0.04mm/meter per division and they work out to be around 0.00240 degrees.
As above the divisions can be split into 5 pretty easy by eye which then works out to be around 0.00048 degrees or better.

So a digital level/angle gauge that has an accuracy of 0.1 degrees is over twice what a standard builders level is, and in my opinion is not really suited to machine work. It would need to have an accuracy (not resolution) of at least 0.01 degrees to be any good to use to set up a lathe etc. Even then it is a the high end of the scale because you can not spit 0.1 - 0.2 degrees etc, where a vial level can be split by eye between graduations.

The trig calculator that I used is here
http://www.pagetutor.com/trigcalc/trig.html

And the metric to anything calculator is here
http://www.convert-me.com/en/convert/length


Dave

[jim]

easy thing to remember with angles is-

one degree over one inch = .0175"

so .1 degree error over 1" = .00175"

[Rob.Wilson]

Interesting Dave 


Thats a handy trig calculator   

Rob 

[foozer]

Like that calc, once I cleared up the mix of feet and inches a 0.100 change in height over 18 feet came out to an angle of 0.030 or 0.001 degree per foot.  Way to small a number for anything I need to worry about, but interesting none-the-less

Robert

[Davo J]

That sounds simple Jim, is their an easy one like that for metric?

Rob
Thanks for that, I would hate to work it out on paper. LOL
I found that one when I was making a chart for all the Morse tapers and NT/INT tapers in degrees for setting up the taper function in the DRO for the lathe. It's just a shame it wont store them in the DRO.


Dave

[mklotz]

That sounds simple Jim, is their an easy one like that for metric?

For very small angles the tangent (and sine) of the angle are equal to the angle expressed in radians.

Thus:

tan( 1 deg) ~= 1 deg * (pi radians/180 deg) = 1 * 0.0175 = 0.0175 radians

The height divided by the length is equal to the tangent of the angle.  Thus we can find the height corresponding to a given small angle by multiplying the tangent of the angle by the length.

In mathematical notation this looks like

tan(A) = h/L    so     h = L*tan(A)

Pick whatever metric value of "L" is convenient.  Let's say, for example, L = 100 mm.

Then:

h = 100 * tan(A) = 100 * 0.0175 = 1.75 mm

so a one degree error corresponds to 1.75 mm displacement over a length of 100 mm.  Equivalently, a tenth degree error is 0.175 mm over 100 mm.

Plug in whatever value for L you find convenient.