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Illustration 3.2: Motion on an Incline

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Galileo was the first person to realize that a well-polished (a very slippery or frictionless) inclined plane could be used to reduce the effect of gravity. He realized that if you started with a vertical incline (angle of 90°), the 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. 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 the incline's angle:

geff = g sin(θ),

where geff is the acceleration down the incline. See Illustration 2.5 and Chapter 4 for more details.

By varying the type of object sliding down the slippery (frictionless) incline, he was able to show that all objects accelerate at the same rate. Try the experiment for yourself (time is given in seconds and distance is given 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 objects' successive displacements increased as odd integers: 1, 3, 5, 7, .... 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. Does that idea look familiar? It should. We found in Chapter 2 that x = x0 + v0*t + 0.5*a*t2 or that, for no initial velocity, x = x0 + 0.5*a*t2 or that Δx is proportional to t2.

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