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Experiment 2Galileo's Experiment - Exploring Accelerated Motion

Pre-Lab Assignment

This three part activity will use three different methods to experimentally determine theacceleration due to gravity.

Learning Objectives1. Linearize a graph to determine the exponent of a parabolic relationship.2. Calculate the acceleration of an object given displacement versus time data.3. Determine the acceleration due to gravity directly by measuring the time it takes

an object to fall a known vertical distance.4. Measure the displacement versus time of an object under the influence of gravity,

but not in free fall, and use the data to determine the acceleration due to gravity.5. Use a motion sensor to measure the displacement of an object versus time, and

use this data to determine the acceleration due to gravity.

Pre-Lab Questions(Complete this assignment prior to beginning the activity)

Prior 10 Galilee, most people subscribed 10 the Aristotle vielt' of falling objects, believingthat the heavier an object, the faster itfalls. Galileo however proved otherwise. Folkloreclaims that Galileo dropped various size spheres from the top of the tower of Pisa.Although this account is a fable, it does illustrate Galileo 's beliefthat all bodies fallunder the influence of gravity at the same rate of increase in velocity. We nmv know thatthis rate is 9.81 m/s', but in Galileo's time no one had yet determined the 'value. Theevolution of our understanding of falling bodies originated with a ver:.vimportantexperiment that Galileo performed. He realized that he couldn 't simply drop an objectand measure how long it takes to fall- primarily because in his time there were nostopwatches or even a clock that could measure time in seconds. Gali/eo realized that ifhe lvere to measure the acceleration of a falling body, he would have to reduce the effectofgravity and thus slow down the "falling" body. Galileo did this by rolling spheresdown an inclined plane. He studied the displacement versus time and the time versusdisplacement and deduced that the distances traversed by a sphere down the incline areproportional to the ratio of the square of the time: dcc t'. Furthermore, he found that thetime it took an object to roll dOH!l1 an incline was independent of its mass, thus provinghis belief that all bodies do indeedfall at the same rate. 111 this case, the rate is thechange in velocity.

-:-c:-:---=---=-----=---:-----c~------: .. .--.---.--------Copyright ,~,2010 by Borislaw Bilash II and VJVREducation 25'1

Purchaser of this kif is allowed to make up /0 100 copies of these instructions per school year.

1) Search on the Internet and to find out more about Galilee' s experiment with inclinedplanes. Specifically, find out how he solved the problem of measuring time without astopwatch. Outline his solution in the space below.

Experimental data may be used to make predictions if you can determine a pattern orthe relationship in the data. One lvay ofdetermining a relationship is 10 pial a graphof the data. 177eresulting graph will give you an idea how two variables are related10 one another. Most of the relationships encountered in this course are linear,parabolic, or inverse.

v / Y Y ( y

x x x x

Linear Parabolic Parabolic Inversev=mx y=mxn vll=mx y=m/x

The simplest of these relationships is the linear relationship which results in astraight line. 17w relationship between the 111'0 variables is summarized by theequation Y = mx + b, where b is the y-intercept.

Variables that are related exponentially (parabolicly) have a more complexrelationship that can be simplified by "linearizing" the graph of the data. Linearizinginvolves modifying the data so that it may be plotted as a straight line.

2S2Copyright {.' 2010 by Borislaw Bilash 1Iand VIFR Education

For example, suppose one of your experiments yields the following data:

x y

1 22 12

3 41

4 965 188

2) Plot the data in the grid below.

The graph you have plotted above is obviously not straight. It appears to beparabolic in nature. This means that y is proportional to x raised to some power-perhaps something as simple as raised to the power of 2 or 3.

Let us guess that J! = 171_/. To check our guess H'emust plot y versus x2 to determine if)

the line is linear. fry versus x~ is infact linear, then we have determined the correctrelationship.

Copyright [) l010 bv Borislaw Bilash 11and VTVR Education 2S3

2 23) Calculate x and then plot y versus x .

2X X Y

1 22 12

3 41

4 965 188

Notice that the graph ofy versus X2 does not produce a straight line, indicating thatourfirst guess lvas incorrect.

Copyright C 2010 bv Borislaw Bilash 11and VWR Education 2S4

This time we will f::,'7lessthat y = mx' and plot y versus x3.

4) Calculate x3 and then plot y vs x3

X x3 y

1 22 123 41

4 965 188

5) Calculate the slope of the line in Question 4.

6) State the equation of the line in terms of y=mx".

The acceleration of an object may be calculated by analyzing the displacement versustime data ofa moving object. To do so, you must first calculate the change in velocitybetween each x versus f data pair.

Copyright 'Q 1010 bv Borislaw Bilash II and VJf!R Education 2S5

7) The data below summarizes the displacement versus time of an accelerating object.Calculate the velocity between each data pair. (Notice that ten x vs t data pairs resultsin only nine velocities).

Data Pair t (s) x (m) Velocity (m/s) Acceleration (rn/s")1 1 1.88

2 2 7.503 3 16.884 4 30.005 5 46.886 6 67.507 7 91.888 8 120.009 9 151.88 N/A10 10 187.50 N/A N/A

The acceleration of an object is equal to the change in velocity per unit time:

Acceleration = (V2-. Vl)l(t~ - t.)

8) Calculate the acceleration of the object described above and record your answers inthe table.

9) Read the Student Guide prior to corning to class to conduct your experiment.

2.)'6Copyright '9 J010 bv Borislaw Bilash 11and VJYR Education