## Motion on an Incline

#### (Illustration)

 Incline 1 Incline 2 Incline 3

Galileo was the first to realize that an inclined plane could be used to reduce the effect of gravity.  His breakthrough was that he realized that if you started with a vertical incline (angle of 90 degrees) this scenario was equivalent to free fall.  If the incline was horizontal (an angle of  0°), the object would not move at all.  He therefore reasoned that as you decreased this angle from 90° the acceleration would decrease and he was able to measure this acceleration, and thereby determine the acceleration due to gravity.  Mathematically, this amounts to the realization that as a function of incline angle

geff = g sin(θ),

where  geff is the acceleration down the incline.  See Illustration 1.4 in the Physlet Workbook for more details.

By varying the types objects rolling down the incline, he was able to show that all objects accelerate at the same rate.  Try the experiment out for yourself (time is measured in seconds and that distance is measured in meters) with the above three animations.

Galileo started his objects from rest on an incline.  What did he find from his experiments?  Galileo's conclusion was that during successive equal time intervals the object's successive displacements increased as odd integers: 1, 3, 5, 7, ...  But what does that really mean?  Consider the chart below which converts Galileo's data into data we can more easily understand (data shown for an incline whose angle yields an acceleration of 2m/s2):

 elapsed time (s) displacement during the time interval (m) total displacement (m) 1 1 1 2 3 4 3 5 9 4 7 16

The third column is constructed by adding up all of the previous displacements that occurred in each time interval to get the net or total displacement that occurred so far.  What is the relationship between displacement and time? The displacement is related to the square of the time elapsed.