Graphical Analysis IIA The Acceleration Due to Gravity PSC 151 Laboratory Activity 3 Graphical Analysis IIA Nonlinear Graphs 1 and The Acceleration Due to Gravity

Graphical Analysis Exercise Determining the Relationship between Circumference and Diameter Procedure: 1. Measure the Circumference and diameter of five circular objects. 2. Analyze data using graphical analysis.

Plot a graph of Circumference versus diameter.

Slope has NO units CALCULATIONS AND OBSERVATIONS: 1. Is your graph a straight line? YES 2. Does the graph pass through the origin? YES…b = 0 3. Are circumference and diameter directly proportional? YES 4. Calculate the slope; Points Used: (4.8cm,15.4cm) & (11.5cm,36.2cm) Slope has NO units

What is the equation relating Circumference and diameter? Compare slope = 3.1 to p (=3.14)

Non-Linear Graphs

We measure its distance each second for 10s. What procedure do we follow if our graph is not a straight line? Consider an experiment designed to investigate the motion of an object. We want to determine the relationship between the object’s distance traveled and time. We measure its distance each second for 10s. Here is the resulting data.

Data We then plot a graph of distance versus time.

Not a straight line but is a uniform curve

Compare graph to graphs of other functions of the independent variable

Plot a new graph where time squared is the independent variable: Distance, d versus Time Squared, t2

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Analysis of Graph

With units of m/s2 the slope represents the acceleration of the object.

It will be difficult to determine the intercept from the graph!

Two Other Methods for Determining the Intercept 1. The intercept is the value of the dependent variable where the graph intersects the vertical axis. At this point the value of the independent variable is zero. Look at the data table to determine the value of d where t2 equals zero. 2. Start with the partial equation: Solve for “b”: Choose any data pair and substitute the values of “d” and “t2” into the equation for “b”: (25s2, 127.5m)

Final Equation

Graphical Analysis of “Free-Fall” Motion Determining the Acceleration Due to Gravity

acceleration due to gravity, g. Purpose: In this lab, you will determine the correct description of free-fall motion and to measure the value of the acceleration due to gravity, g. Introduction: The Greek natural philosopher Aristotle was one of the first to attempt a “natural” description of an object undergoing free-fall motion. Aristotle believed that objects moved according to their composition of four elements, earth, water, air, and fire. Each of these elements had a natural position with earth at the bottom, then water, then air, and fire at the top. If a rock, composed primarily of earth, was held in the air and then released its composition would cause it to return to the earth. Accordingly, Aristotle thought that objects fell with a constant speed which was proportional to the object's weight, that is, a heavier object would fall faster than a lighter one.

Motion at a constant speed can be described by the equation: where d is the distance fallen, v is the speed, t is the time the object has been falling, and d1 is the initial distance from the origin. Comparing the equation above with the slope-intercept equation of a straight line, Y = mX + b, we see that a graph of distance fallen versus time should be a straight line with di as the y-intercept, and the slope of the line would give the speed, v, at which the object was falling.

In the late 16th and early 17th centuries Galileo challenged much of the work of Aristotle. Working with objects rolling down inclined planes he demonstrated that objects fall with a constant acceleration that is independent of their weight. According to Galileo objects fell with a speed that changed uniformly and at the same rate for all objects. Motion at a constant acceleration, starting from rest, can be described by the equation: where d is the distance fallen, a is the acceleration, t is the time the object has been falling, and di is the initial distance from the origin.

Comparing the equation above with the slope-intercept equation of a straight line, Y = mX + b, we see that a graph of distance fallen versus time squared should be a straight line with d1 as the y-intercept and the slope of the line would equal one-half of the acceleration at which the object was falling.

To find the true nature of Free-Fall: Let a ball roll down an incline, Measure the distance traveled after certain times, Plot graphs of distance versus time and distance versus time-squared. If distance versus time is a straight line then Free-Fall is at a constant velocity and the slope of the graph measures that velocity. v, velocity = m, slope If distance versus time-squared is a straight line then Free-Fall is at a constant acceleration and the slope of the graph measures one-half of that acceleration. a, acceleration = 2m, 2 x slope

The Acceleration Due to Gravity If distance versus time-squared is a straight line graph then Free-Fall is at a constant acceleration and the slope of the graph measures one-half of that acceleration. The acceleration, a, found from the slope of the d vs t2 graph is related to but not equal to the acceleration due to gravity, g. To find the actual value of g we must account for the effect of the incline. If the length of the incline is L and the height of its raised end is h:

Experimental Procedure

The metal ball will be released from rest at designated positions (1-5) along the inclined track and the time to travel to the bottom measured with a stopwatch.

distance versus time squared Plot graphs of: distance versus time and distance versus time squared