ACCELERATION DUE TO GRAVITY

Knight Foundation Summer Institute

Elizabeth Chesick, Haverford College

Introduction:

To introduce the concept of "Acceleration due to gravity", drop two objects. Ask the students which object will reach the ground first (i.e. which will fall faster?) In reality, both objects will reach the ground at the same time since all objects accelerate at the same rate. The value of this acceleration is known as "g". (Note: We must ignore air resistance here. For example, if a feather and a brick are dropped at the same time, then the brick will reach the ground first since it is less susceptible to air resistance.) We’ll discuss "g" further below but we must understand a few concepts -–for example, mass:

The mass of an object is a measure of its resistance to being put in motion or, if moving, its resistance to being stopped. Mass is measured by comparison with a standard, such as a standard mass of value measured in grams or kilograms. The comparison is made by using an equal arm balance. The object of unknown mass is placed on the left pan and the standard masses are placed on the right side. Masses are added until the balance is in equilibrium. The mass is then measured in the metric system in grams, kilograms, or milligrams. (Mass in the English system is in slugs). The value of the unknown mass is the value of the standard masses on the right side. So, mass does not change, no matter where it is put because the size of the mass will always be determined by comparison with standards on an equal arm balance.

Weight, however, is the pull of gravity on an object. Weight changes depending on the location of the object. It can be found by hanging it from a spring balance. Weight, is a force and in the metric system is measured in Newtons. In the English system, weight is measured in pounds. Weight is related to mass, in fact it is proportional to mass by:

Wt = mg where g is a constant

Using Newton’s Second Law, F = ma and the fact that weight is a force,

mg = ma

The mass, m, cancels out so that g equals a, or all objects have the same value of acceleration to the ground. If there is no air resistance, (objects are in a vacuum, or are on the moon, or are very heavy with little surface area) all objects reach the ground at the same time. For most places on earth, "g" is very close to 10 m/sec2 (9.8 m/sec2). What does that mean? It means that at the end of one second, the velocity of the object is 10 m/sec, at the end of 2 seconds, the velocity is 20 m/sec. At the end of 3 seconds, the velocity is 30 m/sec. Or, every second adds 10 m/sec2 to the velocity. Or

v = gt.

To be very precise, use 9.8 instead of 10 m/sec2.

What is the relationship for the distance covered? The object goes a certain distance in the first second. In the second second, the distance covered is more because the object is travelling faster, and even more distance covered in the third second since the object is travelling even faster in the third second. Using real numbers, after the first second the object has covered 5 m (or 4.9 m) since the final velocity is 10 m/sec and the original velocity 0 m/sec for an average of 5m/sec. This was determined using the equation:

v = d/t or d = vt

In the second second, the object has an average velocity of 15 m/sec (beginning 10m/sec, final 20 m/sec) so the distance covered is 15m. From the beginning the distance covered is 20 m. For the third second, the average velocity is 25 m/sec (beginning at 20 m/sec and final at 30 m/sec) for a distance covered of 25 m. Or from the start, 5 m + 15 m + 25m = 45 m. The algebraic relationship is:

d = ¸gt2

Substituting "g" = 10 and "t" = 3 gives the answer of 45 m. For more precision, use 9.8 instead of 10 for "g’. "g" can be found by the equation g = 2d/t2

Even though "g" does vary at different positions on the earth, it possible to measure it. In fact, the following lesson allows students to measure the value of "g" using several different methods.

Objectives:

1. To measure the value of "g" by dropping an object out of the window.
2. To measure the value of "g" by using the white paper tape and the timer.
3. To measure the value of "g" by using a laser and computer.
4. To measure the value of "g" by using a super ball and meter stick.
5. To measure the value of "g" by using the pendulum.

Vocabulary:

acceleration due to gravity

distance

velocity

slope

period

Materials:

for a group of 2-3 students

• clay object to drop out of window
• digital timer
• string
• meter stick
• bell clapper timer
• ring stand
• white paper tape
• object to drop through timer
• laser and computer with software
• super ball
• pendulum and clamp

Procedure:

1. Dropping an object out of a second or higher story window
1. Locate a window on the second or third floor from which one can drop an object.
2. Measure the distance to the ground by using the string and meter sticks.
3. Give the digital timer to someone standing at the bottom, where the object will hit. the ground.
4. Drop the object out of the window several times. In each case, start the timer when the object is released and stop the timer when the object hits ground.
5. Keep track of the time in a table in the lab book in a chart such as below.
6. Calculate the value of "g" for each run. "g" should be 980 cm/sec2.
 Trial (distance) d (time) (acceleration due to gravity) g 1 2 3 Etc.

There are a number of errors associated with this method. Measuring the distance is difficult enough to introduce inaccuracies. Also, the digital timer may not be started and stopped precisely enough.

2. Dropping an object attached to paper tape through a bell clapper timer.
3.  Set up the timer on a ring stand so that an object can pull the paper tape through the times as the object falls to the ground. Attach an object to the end of a piece of white paper tape. Thread the tape through the timer. Let the object go. Make several tapes. Analyze the tape by dividing the dots into intervals of 6 dots each.(6 dots corresponds to 0.1 second.) Put the numbers into a chart. Make a graph of length of interval vs number of interval (length = velocity, interval number = tenth of second) Connect the points with the best straight line. Find the slope. The slope is the acceleration - in this case, due to gravity. It should be close to 980 cm/sec2

There are a number of errors with this method also. The paper tape moving through the timer introduces so much friction that the value is likely to be closer to 700 cm/sec2 than 980 cm/sec2.

4. Using a laser and a computer.
5. There exists software to use with a computer and a laser to measure the value of "g". The value obtained b y this method is very accurate. However, everything happens so fast and very little of the physical setup is visible so that the students do not know what happened. Thus, this method is not recommended.

6. Dropping a super ball from a height
1. Obtain a super ball, meter stick, and digital timer.
2. One partner should hold the meter stick vertically on the table with the "10" end touching the table.
3. That same partner should drop the super ball from the "100" end of the meter stick so that it will bounce on the table.
4. When the ball hits the table, the digital timer should be started by the partner who is responsible for timing.
5. The height that the ball bounces to must be noted.
6. When the ball hits the table again, the timer must be stopped.
7. Make three (3) measurements.
8. Enter the data into a table such as the one following. The time showing on the timer is twice the time for the ball to fall. Divide the time on the timer by 2 to get "t". "d" is the height the ball bounces to. "g" is the "g" in d = 1/2 g. Therefore g = 2d / t2

 Trial time on timer t d g 1 2 3

This method should give reasonable results. Probably one trial of the three will be close.

1. Using a pendulum
1. Obtain a ring stand, pendulum clamp, and mass on the end of a string. Assemble the equipment as shown.
2. Measure the length of the pendulum
3. Allow the pendulum to swing back and forth 10 times and measure the time. The period, T, for the pendulum is a tenth of this value.
4. See Pendulum experiment for details.
5. Make a graph of T versus L. Connect the points in a straight line.
6. Find the slope of the line. The slope may be used to find "g". The value of "g" will vary at different places on the earth.

This method does give excellent results and does show that the value of "g" may be found in the lab.

Assessments

-Have the students turn in their data tables, graphs and calculations for "g" to show that they understand the formulas. Make sure they include the reasons for their errors. Some examples for errors are measuring the length, operating the timer, doing the calculations, counting and graphing.

Extensions

See the lab "The Pendulum" in this booklet for a great experiment on what makes a pendulum swing at a certain rate. Knowledge of "g" is very helpful for this experiment. "The Pendulum" also gives the students a great opportunity to design their own experiment!

Another lab dealing with the effects of forces in this booklet is called "Newton’s Second Law – Changes in velocity with Constant Force". Check it out!

To bring the concept of gravity into the students’ own lives, ask them to write a story about what life would be like without gravity. What would the world look like? How would our daily activities change? How would sports change? Would anything stay the same? Which would you prefer – life with gravity – or life without gravity? Use your imagination and have fun thinking about this new world…

Philadelphia Science Content Standards:

-Science Content Standard #1: Nature of Science

This experiment satisfies Benchmark #3 for grades 5-8:"Collect and summarize data from an experiment and interpret the results in terms of the data."

-Science Content Standard #2:Physical Setting

This experiment satisfies Benchmark #4 for grades 5-8: Investigate the relationship between force and motion."

Cross Reference

Although this lab is primarily a physics lesson, it incorporates many skills from other subjects. One example is math. This lab provides a lot of practice in measuring, graphing and calculating. The lab also gives the students a lot of practice collecting data in organized data tables. Next, this lab is a social one; the students are always busy either holding an object, timing, or measuring as they work together and help one another. Finally, the "Extensions" section provides an opportunity to practice writing skills and be creative.

Value of "g" Acceleration due to gravity at different locations

 Place Latitude Altitude "g" in m/s2 North Pole 90o 0m 9.832 Green Land 70o 20m 9.825 Stockholm 59o 45m 9.818 Brussels 51o 102m 9.811 Benff 51o 1376m 9.808 New York 41o 38m 9.803 Chicago 42o 182m 9.803 Denver 40o 1638m 9.796 San Francisco 38o 114m 9.800 Canal Zone 9o 6m 9.782 Java 6oSouth 7m 9.782 New Zealand 37oSouth 3m 9.800