Galilei and the accelerated motion
 

 

Click for IYA

 

 

The visit

The 2009 was declared by the  United Nations the International Year of Astronomy because it coincides with the 400th anniversary of Galileo’s first observations with an astronomical telescope in Padua.

For the occasion the exhibition “Galileo’s future” is taking place in Padua, aiming to show how Galileo’s intuitions gave the basis for many up to date inventions, from the more sophisticated telescope to the nanotechnology.

The 2°E class of our school visited  this exhibition and had the opportunity to observe ancient instruments and equipment, such as an old woody inclined plane.

The experiment in the class


Back in the class, we replicated Galileo’s experiment which lead to the law of falling bodies.


First of all we read the Galileo’s words

(from Discourses and Mathematical Demonstrations Concerning Two New Sciences, 'Third Day')


…. A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball.

Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent.

We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat.

Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former.

Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i. e., of the channel, along which we rolled the ball.

Our experiment

Following  Galileo’s explanations, we prepared our inclined plane.


Materials: an U shaped aluminium (2 m), a little steel ball, books to support the bar, chronometer


Procedure

  • Mark along the bar distance intervals of 0,25 m, starting from an edge
  • Elevate an edge of the metal bar, using a book pile as support
  • Clean the inner bar surface
  • Put the little ball inside the bar at 0,25 m from the lower edge,  let the ball roll along the plan and measure the time. Replicate this fall five times, recording the time taken for each fall
  • Repeat the little ball fall from different distance (0,5 m, 1 m, 1,5 m…), taking 5 measures of time for each distance
  • Calculate the arithmetic mean of the measures from each distance
  • Plot the data into a graph (distance vs. time)
  • Calculate the square mean time and draw a second graph, distance vs. squared time

We replicated the experiments exploring the influence of these variables on the time:

  • Inclination of the bar; the more inclined the bar is, the greater the acceleration is. When, at the limit, the inclination is 90°, the acceleration value is 9,8 m/s^2
  • Mass of the small ball (we used a marble instead of a steel little ball). The mass does not affect the fall time.

 

Fall along an inclined plane

 

 

 

 

 

 t (s)

s (m)

t^2

s/t

s/t^2

1,3

0,25

1,7

0,19

0,15

1,9

0,5

3,6

0,26

0,14

2,6

1

6,8

0,38

0,15

3,2

1,5

10,2

0,47

0,15

3,6

2

13,0

0,56

0,15

                                          Little ball mass: 16,2 g     Slope: 3,5°

s/t is not a constant, while the values of s/t^2 don't change, if we consider the error range.

Experiment with the motion detector

High of the rail above the table: 3 cm  rail length:   1,1 m

The function that better fits the experimental curve is:

The coefficient A of the parabola is ½ a, where a is the acceleration.
The c value (0,538) represent the starting distance of the marble from the detector.

If this distance = 0

the law is:     

The second graph (velocity vs.time) is linear, because during the fall the acceleration is constant. The slope represent this acceleration.