### SC 130 Physical Science Significant Digits Laboratory

#### Set Up

Obtain a meter stick and a cube or a sphere.  Both meter sticks and spheres are limited in number, so some groups will have to start with the cube while others will have to start with the sphere.

#### Experimental Procedure: Cube

1. Measure the Length, Height, and Width of the cube to an accuracy of one millimeter.

Length ______________ Width _____________ Height ______________
2. Calculate the volume of the cube by multiplying the Length times the Width times the Height.

Volume ___________________  (be sure to include the correct units!)
3. Record the Volume on the white board at the front of the room.
4. When every group has recorded their volume, note the largest and the smallest volume shown on the board.

Largest volume ________________ Smallest volume ________________
5. Calculate the average volume by adding all of the volumes on the board and then dividing by the number of measurements  of volume made:

Average  =  Sum of the volumes/Number of measurements = _______________
6. Make the following calculations:
1. Upper half range = Largest volume - Average volume = _____________
2. Lower Half range = Average volume - Smallest volume ______________
7. Fill in the blanks to record the experimentally determined Volume of the cube:

_____________    _____________    _______________
Average volume + Upper half range - Lower Half Range

#### Experimental Procedure: Sphere

1. Measure the radius of the sphere to an accuracy of one millimeter.

2. Calculate the volume of the sphere using the formula for the volume of a sphere:

Volume sphere = ( 4 p r³)/3  where p is taken from a calculator or is 3.14

Volume ___________________  (be sure to include the correct units!)
3. Record the Volume on the white board at the front of the room.
4. When every group has recorded their volume, note the largest and the smallest volume shown on the board.

Largest volume _____________ Smallest volume ______________
5. Calculate the average volume by adding all of the volumes on the board and then dividing by the number of measurements  of volume made:

Average volume =  Sum of the volumes/Number of volume measurements = _______________
6. Make the following calculations:
1. Upper half range = Largest volume - Average volume = _____________
2. Lower Half range = Average volume - Smallest volume ______________
7. Fill in the blanks to record the experimentally determined Volume of the sphere:

_____________    _____________    _______________
Average volume + Upper half range - Lower Half Range

#### Theory

Suppose a cube has a Length L with an uncertainty e in that length measurement.   The Length of one side is therefore L ± e.  The volume is equal to (L ± e)³.  [Expand this binomial as a group exercise.]

The expansion is L³ ± L²e + L e² ± e³.  Note e² is always positive and therefore always tends to increase the volume.  The term that causes the most "problems" in terms of accuracy is L²e.  L²e is an uncertainty in the area due to uncertainty in the length of the edges of the area.  This term can be quite large.  If a cube is five centimeters on an edge and the uncertainty is one millimeter, then this term is plus or minus 2.5 cm³.  Similar arguments can be made for any volume calculation, there is always an areal error that is "large" compared to the error in measurement of the length of a edge.

These same arguments apply to the calculations of the volume of the sphere.

#### Application: Significant Digits [Supplemented by lecture at board]

When adding numbers simply add the numbers and keep the result.  This is not wholly correct, but without being able to calculate an actual experimental error in each measurement this procedure will have to suffice.

When multiplying numbers, count the number of digits in each measurements.  Keep the number of significant digits that is equal to the number of the digits in the measurement with the fewest digits.

5.4 cm + 11.9 cm  + 101.5 cm = 118.8 cm

Often, when adding, the number of decimal places in the measurements will be the same.   In this case the number of decimal places in the answer agrees with the number of decimal places in the answer.

5.4 cm × 11.9 cm  × 101.5 cm = 6522.39 cm³ = 6500 cm³

Note that only the 6 and the 5 were kept.  The number had to be rounded off to the nearest hundreds place to keep only two significant digits.  The measurement that limited the problem to two significant digits was 5.4 cm.

Remember that zeros are not counted as significant digits when they are to the left of the decimal place.

Record the volume of the cube and the sphere to the correct significant digits, do not forget to include the units!

Cube ___________________            Sphere: _________________