### 1.8M mirror mass

The 1.8M LAT Mirror is estimated to weigh 2420 lbs. ***(+/- 128 lbs.)

The mass of the glass was determined in the following matter.

(At the moment we are not able to weigh the glass directly.) Therefore.
We need to know the density of the glass and the volume of the glass.
As it turns out. We have some chips that were accidently knocked out from the mirror blank.

• First the shard was weighed. mass = 16.000 gm
• Then the volume was determined. Two techniques were used.

Archimedes water displacement (overflow) technique.
Buoyancy technique. (sample weight is less its displacement in water.

Shard volume = 7.1 cm^3

• The density of the glass is (mass/vol.)= 2.253 gm/cm^3 (+/-0.027)
• Now we need to know the volume of the mirror blank, which is the volume of the cylinder minus the volume which has been removed to form the speculum.
• Diameter of the blank is = 180 cm
• Thickness is (approximately) = 21.2 cm
• This give us a volume = 539,500 cm^3.
• We can find the volume of the glass removed to form the speculum, from the formula for volume of a spherical segment which is

1/3*pi*h^2*(3r-h)

We need 2 more measurements for this.

• h= sagitta = 4.064 cm (depth of the speculum)
• r= the radius of curvature = 1009.0cm
• The volume of the glass removed to form speculum = 52,880cm^3
• Corrected volume =487,200 cm^3
• Corrected mass = 1.098E6 gm. = 2420 lbs.

** speculum; The part of the glass which is "silvered" the "actual" mirror.
** sagitta; The depth of the speculum. (our mirror the sagitta is 1.6 in.)
***error(s);The most sensitive measurements to total mass error is the thickness. (+/- 1 mm.) The next most sensitive measurement (due to its poor quality) is the volume of the shard; which gives us a density error of +/-0.27. Assuming that all the error (>90%) are in these two measurements, gives a total weight error of +/- 128lbs

Mark Thein did more work.

Rick:

I calculated the volume of the glass by using the Volumes-by-Cylindrical-Shells method of integration.

First I came up with the formula for the cross-section of the glass with the curve as a parabola. The equation of the parabola is Y=X^2/(4*fl)+ T with fl being the focal length in cm and T being the thickness of the glass at the center, which is 17.3cm.

Next I plugged the formula into the volume equation V=the integral from 0 to 90 of 2*pi*X*f(x) with 90 being the edge radius in cm.

The volume equation for the glass is then V=(pi*X^4)/(8*fl)+T*pi*X^2.

After placing 17.3 for T, 519.1125 for fl and 90 for X the volume is 489684 cm^3. Multiply by 2.25 to get grams and you get 1.102194*10^6g or 1102.194Kg or 2430lbs.

This result is close to yours. This volume equation can be used for any parabolic mirror with a flat back.

Mark Thein

Rick Powell rick(AT)group70(DOT)org