**Learning objective:**

Recognize the importance of indirect measurements.

Conduct an experiment in the method and effectiveness of indirect measurements.

**Something to think about:**

*The volume of the nucleus of gold atom was determined by "firing**" alpha particles at a sheet of gold. The number "hts" or fired particles was 3.4 x 10^4, 1 in 8000 particles scored a "circle hit".*

**Investigation**

In 1909 Geiger and Marsden performed an experiment to prove the atomic model called "the plum pudding model". Particles were fired at a thin sheet of gold (and other material) with the expectation that they pass right though without changing its path. What they discovered was the 1 out of every 8000 particles fired were reflected back. From this observation they could conclude that the plum pudding model was wrong, and the center of the atom (what we now call the nucleus) was very small compared to the total volume of the atom.

Problem: Determine the radius of a single target circle indirectly.

Procedure: Use copies of either of the circle sets on the following pages. Have students place the circled paper on the floor, face up. Students drop markers so that they hit the paper within the marked boundaries. Repeat this at least 100 times. It may be more convenient to drop the markers from just above the paper; however, one should then take care to distribute the hits as randomly as possible over the entire target area.

Analysis: Count the total number of dots on the paper (hits) as well as the number of dots which are completely within a circle (circle hits). Determine the total area of the paper (rectangular area), and count the total number of circles on the paper. If we have uniform circles and a random distribution of hits, then we can assume:

Therefor you can calculate

Subsequently you can calculate:

The area of one circle can be used to calculate the radius of a circle: area = 3.14(radius)^2.

This calculated radius can then be compared with a direct radius measurement.

Conclusion: Things to think about when it comes to error analysis: How would you determine the uncertainty of your measurement? Do the hits that are "sort of" in count as circle hits or not? How about the number of circles (not all of the circles are complete)? What was your percent error (the radius should be 1 cm, but when printing the aspect ratio may not be the same)