Unit 6: Exploring Gravity

Gravity is one of the most fundamental, and most mysterious, forces in our universe. There is an adage in astronomy that “Gravity controls everything.” Gravity was first explored mathematically and scientifically by Galileo in the early 1600’s. It was Galileo who first realized that gravity acts on all things equally and that everything falls at the same rate regardless of its mass. Galileo also explored gravitation using ramps and pendulums – something that even the youngest students can experience and begin to understand in school today.

The function of gravity on the solar system was largely unknown until Isaac Newton proposed his theory of universal gravitation in 1665. Newton’s theory says that all things possess gravity and attract each other across space. Therefore while the Earth’s gravity attracts you and holds you on the surface, your gravity also attracts the Earth! Newton also proved mathematically that gravitational force controlled all the orbits in our solar system – both those of planets going around the Sun as well as the orbits or moons around various planets. In fact, it was the falling apple that led Newton to prove that the Moon is really falling in its orbit around the Earth. Newton used his ideas to propose that artificial satellites were actually possible some 300 years before anyone actually launched one into Earth orbit.

The concept of what gravity actually is remained mysterious until Albert Einstein figured it out in his Theory of Relativity which was developed between 1905 and 1915. Einstein argued that space and time are actually one unified thing called spacetime. According to Einstein, it was the curvature of this spacetime that really creates the gravitational force that produces the effects studied by Newton and Galileo. While Einstein’s theory is well beyond most of us mathematically speaking, it is perfectly possible for young students to build simple Einsteinian models and explore the concepts of spacetime and gravity in the classroom!

In this unit, the activities we attempt will be arranged historically; that is, we will try some of Galileo’s ideas first, then explore Newton, and finally Einstein. What? You didn’t think you could teach 21st century science to elementary school children!? Yes, you can! Let’s get started!

Activity 14: Galileo Explores Gravity with Pendulums

Legend has it that a young Galileo observed the swinging of a censer in church one day and noted that the incense burners kept swinging in time with each other as long as the chains that held them were of the same length. Galileo constructed his own pendulums and continued to experiment with them for much of his life. Like Galileo, we have much to learn from a swinging weight on the end of a length of string!

Academic Standards

Science and Engineering Practices

  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Argument from evidence.

Crosscutting Concepts

  • Cause and effect.
  • Structure and function.

Next Generation Science Standards

  • Forces and interactions (K-5, 6-8, 9-12).
  • Structure and function (K-5, 6-8, 9-12).
  • Engineering and design (K-5, 6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. Everything has its own gravity and gravity is always attractive. (Newton’s law of gravity.)
  2. Gravity is a property of mass; the more mass an object has, the more powerful its gravitational force will be. (Newton’s law of gravity.)
  3. Gravity makes everything fall at the same rate, but we can reduce gravity’s force, slow it down if you will, by using a ramp or a pendulum. (Galileo)
  4. Gravity is created when massive objects like planets pull on and stretch spacetime. (Einstein’s theory of general relativity.)

Teaching and Pedagogy

K-8 educators often shy away from topics like gravitation because they feel like the mathematics required will be beyond their younger student’s grasp. While this may be true, there is no reason to avoid topics like gravity which can be explored conceptually with low-cost, hands-on activities.

The key to making such activities successful in the classroom is an active guidance by the teacher that points out key ideas. Students, like everyone else, often see things all the while missing the important facts and ideas. Pointing out these ideas is the teacher’s role here; active guidance also helps students avoid forming misconceptions as they explore new activities.

Linking accurate observations with key ideas and explanations is a critical role for the educator. As a classroom teacher, you don’t need a mathematically sophisticated understanding of astronomy and physics to conduct successful science activities. You do need to understand a few key facts and ideas, and be able to recognize them and point them out as your students explore and learn.

Student Outcomes

What will the student discover?

  1. How heavy something makes no difference in how fast it falls. This was very confusing to ancient scientists, and still is to many modern people as well! Aristotle taught that the heavier a thing is, the faster it falls. It was Galileo who first proved Aristotle wrong and showed that all things fall at the same rate.
  2. It is the length of the string, not the weight on the end of it which controls how long it takes for a pendulum to swing back and forth.
  3. The way a weight swings on a string is intimately connected with the way that all things fall. We will explore this further in future activities.

What will your students learn about science?

  1. Old and established ideas are not necessarily true. We do not owe an idea respect or reverence simply because it is old, but we don’t just throw out ideas because they are old, either! Experimental evidence always trumps tradition.
  2. There are often subtle connections in science. The idea that a weight on a pendulum string is actually falling as it swings is both powerful and subtle. Scientists and mathematicians had used and observed pendulums for thousands of years before Galileo discovered this important connection.
  3. Often, we can see and describe a pattern before we can understand it mathematically. In this case, our students will be exploring gravity in a powerful conceptual way to prepare them for the mathematical explanations they will be exposed to in years to come.

Conducting the Activity

Materials

  1. A ball of string, sturdy thread, or twine
  2. 20+ 3/8” metal washers to serve as weights
  3. Several large paper clips
  4. Wooden or plastic ruler
  5. A timer (a stopwatch app on a smart phone works well!)
  6. An 18-inch long flat board. This can be almost anything from a piece of sturdy cardboard, to a 2×4, a piece of shelving…
  7. [Optional] Three cup-hooks (the sort that hold up coffee cups beneath a shelf)
  8. Several rocks of various sizes, from fingernail size to chicken-egg sized.

Building the Pendulum Model:

  1. [Teacher] Securely screw three cup hooks into one side of your flat board. If you are using a piece of sturdy cardboard (triple thickness of copy paper box glued together with white glue does well!), you should use hot glue or super glue to make sure the hooks will not slip out! The board can now be placed across the backs of two chairs so that pendulum weights attached to the hooks can swing freely.
  2. Thread some washers onto three paper clips. 2 washers for the pendulum-A, 5 washers for pendulum-B, and 8 washers for pendulum-C. Tie a 2-ft length of string to each paper clip.

Exploring the Pendulum Model

  1. Now ask the class which pendulum will swing the fastest and which the slowest? Ask they to write down their answers and rate them 1-2-3.
  2. Now it is time to test our predictions! Secure all three strings to the hooks on the board. The exact length is not important, but use a ruler to insure that all three strings are the same length. Knot the strings securely and trim off any extra string with a scissors.
  3. With the board held steady, use the ruler to pull back the weights just a few inches and then release them together as the timer starts. Assign students to count how many complete swings out and back each pendulum makes. The timer lets the clock run for 5 seconds and then calls “STOP!”
  4. Note: Do not pull the pendulums back too far – this will interfere with good results. Pulling back 4-inches at most will work best.
  5. Most students will be amazed that the light, medium, and heavy weight pendulums all swing at the same rate! Weight has nothing to do with the speed of a pendulum! More on this in a bit!
  6. Ask your students to figure out a way for three pendulums to swing as different speeds. It probably won’t take long for someone to cry out: “Try different lengths!” Redo the pendulums so they all have the same number of washers (5 works well). Change the string on them so you have one short, one medium, and one long pendulum.
  7. Ask the students to think and record their predictions on a piece of paper again. It won’t take long for them to see that the shortest pendulum races along, while the longest moves at the slowest pace. Length is the crucial factor in pendulum time!

Discussion Questions

  1. What was the most surprising thing you learned about pendulums today?
    • Answer: Most students are amazed that weight has nothing to do with how fast a pendulum swings. Aristotle’s misconception that heavy things must fall faster than lighter ones is still alive and well today!
  2. Why are Grandfather Clocks so tall? (Look up an image of one on the internet!)
    • Answer: The pendulum on a Grandfather clock is about 2-feet long, this isn’t a coincidence! At this size, each swing of the pendulum from one side of the clock to another takes 1 second. The size of the clock is controlled by Earth’s gravity!
  3. How would you have to change the design of a grandfather clock if you were to build one on the Moon?
    • Answer: In the Moon’s 1/6th gravity, things fall much more slowly, and pendulums swing more slowly as well. In order to make a clock that tick-tocked once per second, the pendulum would have to be much shorter. A grandfather clock on the Moon would be only about 18-inches tall!
  4. What is the main thing that pendulums tell us about gravity?
    • Answer: Because a pendulum swings at the same speed no matter how large the weight is, this tells us that everything must fall at the same speed regardless of weight! Galileo referred to this property of matter as inertia. We will come at this idea again in Activity #16.

Supplemental Materials

Going Deeper

History tells us that Galileo first noticed the relationship between pendulum length and period (the time it takes for one swing out and back again) by watching and incense censer being swung back and forth during a church service. Galileo went on to investigate gravity with pendulums, ramps, even by dropping various iron weights off the leaning tower in Pisa, Italy!

Our exploration of gravity and pendulums is a simple (and surprising!), event for students. But why doesn’t weight cause the time of the pendulum to change? Don’t large weights experience more gravitational force than small ones? Shouldn’t the larger gravitational force cause them to go faster than the lighter weights? Surprisingly, the answer is no!

Galileo realized that although larger weights experience more gravitational force, they are also harder to move – Galileo called this property of matter inertia. You can try it yourself with a couple of rocks. Take a small rock, perhaps one inch across and see how far you can throw it. Now choose a larger rock, say 4 inches wide and see how far you can throw that! Your arm is just as strong, but inertia makes it harder to get the large rock moving so you cannot throw it as fast – or as far. If you are leery of trying to have your students throw rocks in school (really!?), have them try and throw a rubber T-ball and a basketball on the athletic field.

For our pendulums, it is much the same. The greater weight of the largest pendulum means it is pulled down with much more gravitational force than the lighter pendulum experiences. But the larger pendulum is also harder to move – the force of gravity and the pendulum weight’s inertia balance out exactly, and so the period of the pendulum remains the same as long as the length is the same. For our grandfather clock, a pendulum length of 50 cm (20 inches) makes the pendulum tick-tock gracefully with a two second period; one second to swing out, and one second to swing back. A grandfather clock pendulum on the Moon would be much shorter, just 8.2 cm long, because the Moon’s gravity is 1/6th that of Earth. On Jupiter, where the gravity is almost 4x that of Earth, a grandfather clock pendulum would have to be 2 meters long, that’s almost 7 feet!

Being an Astronomer

One of Isaac Newton’s great rivals was fellow British scientist Robert Hooke. Newton had perhaps the greatest mind ever for developing far reaching and sophisticated mathematical models and theories, but Robert Hooke was by far the more practical of the two men. Science needs both types!

After Newton had published his theory of universal gravitation that mathematically demonstrated both how and why planets orbit the Sun in neat elliptical pathways, Robert Hooke upstaged Newton at a meeting of the Royal Society (the British Academy of Science and Mathematics) by demonstrating what is called Hooke’s Pendulum, a simple and effective demonstration of orbital motion that you can reproduce in your classroom today.

While Newton had used many pages of complex and sophisticated mathematics using algebra, trigonometry, and calculus proving that planetary orbits were caused by the gravitational attraction of the Sun, Hooke used a simple mechanical model to demonstrate the same effect in seconds using no math at all! Everyone who saw it understood it almost instantly – your students will too!

Being a Scientist

Calculating the period, or cycle time of a pendulum is not terribly difficult; it can be done easily with most any school calculator. It is often interesting, and productive, for the gifted student to grapple with mathematical explanations rather than simply sticking to conceptual ideas.

The formula for the period of a pendulum involves only three numbers, and only one of these needs to be measured. For the purpose of our calculation, we needn’t worry about units. Our answer will automatically come out in seconds.

image
  • We use T to represent the time or period needed for a pendulum to swing all the way out and back again. This is measured in seconds.
  • We use L to measure the length of the pendulum. This is just the length of the string and is measured in meters. If you use measure the string in centimeters, divide your answer by 100 to convert the value into meters.
  • π (or pi) is the ratio of a circles diameter to its circumference. This number never changes: π = 3.14
  • g is used to represent the gravitational constant for the Earth. This is the rate at which things fall when we drop them – it applies to all objects, regardless of size, weight, or shape. g = 9.81 m/s2.
  1. Begin by taking the Length and divide by 9.81. Write this answer down.
  2. Take the square root of the first answer, write this value down.
  3. Multiply the second answer by 2, and then multiply again by 3.14.
  4. This final answer should be the period of your pendulum in seconds. Measure the period of your pendulum with a stop watch and see how close you get!
  5. Let’s try an example. Let’s say your string is 25 cm long (that’s 0.25 meters!). You calculate the period like this:
image

Following Up

Pendulums are used in many kinds of devices, from scientific and time keeping instruments to musical instruments. Besides the types of pendulums that swing on a string, essentially anything that vibrates back and forth can be considered a pendulum.

A guitar string vibrates back and forth when it is plucked, this back and forth motion is much the same as a clock pendulum – and the mathematics that governs the behavior of vibrating strings and pendulums is much the same. The wings of a fly, even the musical note we make when we blow across the mouth of a partially filled bottle are examples of vibrating pendulums in nature. How many can you think of?

Activity 15: Hooke’s Pendulum

Robert Hooke and Isaac Newton were great rivals both in European science and mathematics, as well as in the Royal Society for Science and Mathematics where Newton was president. Both men were fiercely competitive, and jealous of their work and fame. When Newton published his theory of gravity in the book Principia Mathematica, he struggled and failed to develop a simple and convincing demonstration for the mathematical concept that only a center-seeking force (gravity) and the straight line motion of a mass (momentum) are needed to create an orbit. Robert Hooke’s simple pendulum experiment achieved this and was considered a great triumph!

Academic Standards

Science and Engineering Practices

  • Developing and using models.
  • Constructing explanations.
  • Obtain, evaluate, and communicate information.

Crosscutting Concepts

  • Cause and effect.
  • Systems and system models.

Next Generation Science Standards

  • Forces and interactions (K-5, 6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. For any body in orbit such as a moon or planet, the force of gravity always pulls toward the center of the primary body. For example, the Earth’s gravity always pulls the Moon directly toward the center of the Earth.
  2. If you draw a line from the center of the Moon to the center of the Earth, the Moon’s momentum is always perpendicular to this. In other words, the Moon’s momentum would take it off into space on a straight line. This is true for any body in orbit.
  3. It is the combination of straight-line momentum, and a center-seeking gravitational pull that produces a smooth, elliptical orbit. The realization that an orbit required only two things acting on a moon or planet was Robert Hooke’s stroke of genius.

Teaching and Pedagogy

One thing that Galileo noted about a simple pendulum – a weight suspended by a string – is that the pendulum mass always passes under its point of rest. That is, if we hang a weight by a string, and mark the point directly under the unmoving weight, we have found the point of rest for that pendulum. Pull the weight back and release it, the weight will travel straight back and pass over that point, no matter which direction we start from.

You may also notice that if the pendulum is at rest, the string points directly to the center of the Earth. Any time we drop an object, letting it fall straight downward, the object also falls directly toward the center of the Earth. It is this connection between gravity and pendulums that Robert Hooke noticed and later used in his demonstration.

Gravity always pulls objects toward the center. Every object on our planet falls toward the center of the Earth, every planet is pulled toward the center of the Sun. Gravity is a centripetal, or center-seeking force. Pointing out this connection between gravity and pendulums will start your students thinking more deeply about gravity!

This is a fun and simple activity, but it seems to fascinate everyone. Students of all ages love to play with this mechanism and see how circular or how elliptical an orbit they can create! My suggestion to you is: let them play! As we have seen before, playing with scientific models is a wonderful way to build deep cognitive understanding of how Nature works. Our job as teachers is not to limit the play, but to reinforce the intuitive learning and help students acquire the vocabulary and fluency to express what they have learned to others (and on assessments!)

If you have a ‘Back to School’ night or PTA night, even a science fair, this is a great project for students to use to demonstrate what they have learned. The adults who see this will be just as impressed as the children were the first time they saw the demonstration; not because you can make a weight circle a pendulum, but because of the deep links that can be drawn between the pendulum’s elliptical motion and the Moon’s orbit in space!

Student Outcomes

What will the student discover?

  1. Gravity is a centripetal, or center-seeking force.
  2. Gravity’s center-seeking action is at play whether we consider a pendulum, or a moon in orbit around a planet.
  3. Center-seeking gravity and a perpendicular momentum are the only things necessary to produce a smooth planetary orbit.

What will your students learn about science?

  1. Sometimes science progresses not because of great friendships, but because of great rivalries. Isaac Newton and Robert Hooke were bitter rivals who competed with each other almost all their lives.
  2. Hooke’s pendulum is an extremely simple idea. People have wondered for centuries why Newton didn’t think of the idea himself, but sometimes genius can be found in simplicity as much as in complexity.

Conducting the Activity

Materials

  1. A pendulum weight and a string for each student or group of students (see Activity 13).
  2. A piece of paper with a large dot on it (dot stickers, markers or crayons work well).

Building Hooke’s Pendulum

  1. You can use the same pendulum materials you constructed for Activity #14; a board suspended between two chairs or two desks with a cup-hook attached underneath.
  2. Hang your pendulum mass by attaching its string to the hook beneath the board. It is essential for this experiment that the board is sturdy and that it is held firmly in place so that it cannot wobble or move as the pendulum swings.
  3. Place a piece of paper with a central dot on the floor beneath the pendulum – it is best if the pendulum weight hangs no more than a few centimeters above the center dot. Now your apparatus is ready to go.

Exploring Hooke’s Pendulum

  1. Hold the string so the pendulum is motionless with the weight suspended over the dot. Pull the pendulum back a few inches and release the weight carefully.
  2. Notice that whichever direction you pull the weight back, the pendulum always swings straight back toward the dot in the center. Hooke said this ‘modeled gravitational attraction’, that is, like gravity, the pendulum is always pulled toward the center. The Sun’s gravitational pull tugs every planet straight toward the center of the solar system in this way.
  3. Now try something different: pull the pendulum weight back as before and give the weight a little shove toward the side as you release it.
  4. Instead of swinging straight back toward the dot in the center, the weight now orbits around the center. With a little experimentation, the students will find that it is almost impossible to make the weight go around in a perfect circle. Instead, the weight most naturally follows and elliptical path, an off-center oval shape which causes the weight to travel sometimes closer to the center, and then sometimes farther away again.

Discussion Questions

  1. How do we know that the pendulum is always pulled back into the center just like gravity?
    • Answer: Pull the pendulum weight in any direction you wish. Hold it for a moment, then release it; the weight always swings right back toward the center point! If you drop a rock from anywhere on Earth, it will do the same thing – it will fall directly toward the center of the Earth; we call this ‘falling straight down.’
  2. Is it really impossible to have the pendulum weight go in a perfect circle?
    • Answer: No, just very difficult. You must precisely balance the force of your shove (momentum) with the pull toward the center (gravity). Even if you do this, the friction at the top of the pendulum will slow the weight down and cause it to shift into an elliptical motion in just a few seconds.
  3. Is this really the way that gravity and orbits work?
    • Answer: Yes. Among other things, Robert Hooke took the time to prove mathematically that his pendulum and an orbiting moon are mathematically identical!

Supplemental Materials

Going Deeper

Robert Hooke lived at the same time as Isaac Newton, but he is far less well known. Some of Hooke’s work included designing mechanical devices and the use of the microscope to describe accurately small insects, animals, and cells.

Dive into some of Robert Hooke’s work and see how much the little-known man contributed to modern science!

Being a Scientist

The elliptical orbit (oval shaped) is known to be the universal form for all orbiting bodies in the universe. Circular orbits are possible, but this is an unstable arrangement, like balancing a pencil on its point. As a pencil will quickly fall one way or another if balanced on its point, any circular orbit will quickly decay into an elliptical form.

Can you make a circular orbit using a pendulum? How many rotations does it take before the orbit becomes definitely elliptical (oval) in shape? Experiment with Hooke’s pendulum and see what you can find out!

Following Up

Natural orbits are one thing, controlled orbits are another. Do an internet search and see if you can find the flightpath for the Apollo moon missions, one of the Mars rover spacecraft, or the Cassini or Juno space probes. These spacecraft have beautiful and complex orbits, controlled by engines and precise controls either from astronauts or from ground control scientists.

Activity 16:  Galileo’s Falling Bodies

One of the first biographies of Galileo describes his famous experiment, dropping iron balls of different weights from the top of the famous leaning tower of Pisa. Galileo sought to prove that all objects fell at the same speed, regardless of their weight. You will recall from Activity #14 that the pendulums were also unaffected by their weight; the only way to change the timed length of a pendulum’s swing was to change the length of the string that held it.

Aristotle’s scientific model stated that things fell to Earth because the ‘wanted to reach their natural place’, and that the heavier an object was, the faster it would fall. Although it is simplicity itself to do the experiment that Galileo did, Aristotle apparently never did it. Aristotle’s fame was such that no one seriously challenged his assertions for over 2,000 years. Galileo’s experiment shows us the utility of gathering accurate observational data and comparing it to the predictions of scientific models. This is the very mechanism through which science corrects its own errors.

Academic Standards

Science and Engineering Practices

  • Asking questions and defining problems.
  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Argument from evidence

Crosscutting Concepts

  • Cause and effect.
  • Next Generation Science Standards.
  • Forces and interactions (K-5, 6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. All objects on Earth fall at the same rate. This rate is called the acceleration of gravity, on Earth this rate is 9.81 m/sec2. We use the symbol g to represent this value.
  2. Acceleration means that the velocity at which an object moves is changing is a steady way.

a. Acceleration can be either positive (speeding up) or negative (slowing down.)

b. Earth’s acceleration is 9.81 m/sec2. This means you add almost 10 m/s to your speed every second you spend falling. After 3 seconds falling, you are moving over 29 m/s, that’s almost 66 mph!

Teaching and Pedagogy

This experiment, like the pendulum in activity #14, shows us that inertia and gravity are closely linked and balance out exactly whenever an object is falling freely toward the ground. Our old friend Aristotle famously said that heavy objects fall because they want to reach the center of the Earth.

Aristotle hypothesized that a heavier object would ‘want to fall more’, and so it would fall faster and strike the ground first if dropped with a lighter object. Aristotle was so very well respected as a genius and a scientist that his ideas were not questioned for almost 2000 years! “Aristotle said it, so I believe it!” was the attitude of learned men and women for many centuries.

Galileo was one of the first people to actively confront the established view and test these ideas with experiments, in fact he wrote an entire book of do-it-yourself experiments that allowed people to prove for themselves that Aristotle was wrong about many things, including gravity and the design of the solar system.

Challenging the ideas of Aristotle was highly unpopular; Galileo was imprisoned for his bold, scientific explorations and spent the last 12 years of his life in custody. If we teach children anything about science with these experiments, it must be that there are no sacred traditions in science. We can, and must, question everything by putting it to the experimental test. Those who say: “Don’t you believe?” or “Almost everyone agrees, why don’t you!?” are not being true to our scientific heritage!

Student Outcomes

What will the student discover?

  1. Gravity causes every object to fall at the same rate toward the center of the Earth.
  2. Air resistance can powerfully affect the rate at which a light weight or low density object falls.

What will your students learn about science?

  1. There a no sacred ideas in science. No matter who says something is true, no matter how long we have ‘known a fact’, it must stand up to experimental challenge.
  2. Gravity is a powerful and universal force. Gravitation affects everything, pulling it toward the center of mass (usually the center of a planet or star.)
  3. Gravity and inertia are closely linked. It is the precise match between gravity (the pulling force between two objects like the Earth and a rock) and inertia (the resistance to movement or force) that makes the constant acceleration of gravity a reality.

Conducting the Activity

Materials

  1. Flat board about 18 inches long (See Activity 14)
  2. Various rocks, from fingernail size to as big as a chicken egg. (any weighty object will do)
  3. A single sheet of paper crumpled loosely into a ball
  4. A feather or a dried leaf

Building the Galileo’s Freefall Model

This model is simply a flat board laid along the edge of a desk or table. The idea is that two or more objects can be tipped off the board at the same time and allowed to fall.

Exploring the Galileo’s Freefall Model

  1. Set your flat board along the edge of a table top.
  2. Place any two rocks (or other weighty objects) on the edge of the board
  3. Tip the board slowly and allow both objects to fall off. Ask students to watch to see which one hits the ground first?
  4. Try this with various combinations of weights, then try with a weight and a crumpled piece of paper or a dried leaf. (The rock will obviously hit the ground first.)

Discussion Questions

  1. How did Galileo prove Aristotle was incorrect?
    • Answer: Aristotle would have said that the heaviest objects always fall faster and hit the ground first.
  2. Which results would agree with Aristotle? Why do you think this happened?
    • Answer: The slowly falling leaf and rapidly falling rock would seem to support Aristotle’s viewpoint. In fact, it is the resistance of the air that slows the leaf down, the rock is much denser and is not affected as much. There is a famous NASA video showing Apollo 15 astronaut David Scott dropping a metal hammer and a falcon feather on the Moon; in the airless environment, the hammer and feather both fall together perfectly.
  3. What lesson about science does Galileo’s experiment teach us?
    • Answer: Question everything! No scientific theory is so famous or so honored that it should not be questioned. When a theory makes a prediction, we should gather data to see how that prediction holds up!

Supplemental Materials

Going Deeper

Galileo was known for challenging long held beliefs, primarily the ideas of Aristotle. Apart from exploring ideas about gravity, inertia, and friction, Galileo also challenged the idea that the Earth was fixed, or motionless in space.

Aristotle claimed that if a mountain was too big to be moved, then what force could possibly move the entire Earth? Aristotle also knew the approximate size of the Earth (24,000 mile circumference) and realized that the surface of the Earth must be moving approximately 1000 miles per hour if it spun on its axis once a day. Aristotle said such a great speed would cause huge hurricane winds – and since we don’t feel such winds, the Earth must be motionless.

How would you respond to Aristotle’s arguments? How did Galileo do so? Even if you cannot figure out how to counter Aristotle’s arguments, does that mean that he was correct about the Earth being fixed in space?

Being an Astronomer

Gravity controls the orbits of all satellites of the Earth, from the Moon down to the smallest scientific, weather, and communication satellites rocketed into orbit. The best time to see satellites is in the first 2 hours after sunset or the last two hours before dawn.

Go outside on a quiet and clear night and sit in a lawn chair so you can lean back and watch the sky comfortably; a dark place away from streetlights will be very helpful. As your eyes adapt to the dark, you may notice that there are some fainter stars which drift slowly across the sky.

Watch them carefully – if they are blinking, these are most likely airplanes high in the sky, flying from one city to another. If these drifting stars do not blink at all, then they are likely satellites moving silently across the sky as they travel in low Earth orbit!

Being a Scientist

Once you learn to spot satellites, you can use a simple ruler to judge their speed. Take a lawn chair out as you did before, but this time take a ruler marked in centimeters, along with some paper and pencil to record your findings. A parent or a partner will make the job much easier!

Once you spot a satellite moving across the sky, hold your ruler up at arm’s length and start measuring the satellite’s progress. When you begin measuring, tell your partner “Go!” and have them begin timing. Measure the satellite’s progress for 20-30 cm if you can, then tell your partner “Stop!” Record the time and distance travelled by the satellite.

With a ruler at arm’s length, 1 cm is approximately equal to 1 degree of arc. Divide the distance in degrees by the time in seconds to get the speed of the satellite. Because gravity’s force is stronger the closer that you get to the Earth, satellites which are in a lower orbit travel faster! See if you can rank your satellites from lowest to highest in orbit!

Following Up

You might get the impression from these activities that Aristotle’s ideas were all silly or ignorant – nothing could be further from the truth. See if you can research some of Aristotle’s ideas; what did he do to become so famous?

Activity 17: Packard’s Acceleration Ramp

We know that gravity makes everything fall to Earth; everyone has heard the old saying: “what goes up must come down!” But the question remains, how do we fall? When we jump off a diving board into a pool, there is the rush and then the splash, but what is happening to us as we fall? Why do asteroids strike us at such tremendous velocities, upwards of 30,000 miles per hour; how do such large objects get moving so fast? As we shall see, it is gravity which speeds us up as we fall – we call this speeding up: acceleration.

This activity is a simple one, we use a ramp to slow down the fall of a marble so that we can study it more easily. Galileo, and many other scientists and thinkers through the centuries have used ramps to study gravity. This apparatus was originally developed by John Packard, an American high school science teacher in the early 1900’s; you can recreate this simple device in your classroom and learn more about gravity!

Academic Standards

Science and Engineering Practices

  • Developing and using models.
  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Argument from evidence.

Crosscutting Concepts.

  • Cause and effect.
  • Systems and system models.

Next Generation Science Standards

  • Forces and interactions (K-5, 6-8, 9-12).
  • Engineering and design (K-5, 6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. Gravity is a steady and consistent force. Gravity pulls us steadily toward the Earth’s center, and just as it pulls the pendulum toward its center point, gravity also pulls the ball down the ramp.
  2. The horizontal motion of the ball does not affect its vertical motion down the ramp. Try it! If you release a second ball at the same time that the first ball leaves the ruler and let it roll straight down the ramp, both balls will reach the bottom at the same time!
  3. The curved path of the ball is the result of just two things:

a. The ball’s horizontal speed coming off of the ruler.

b. The angle of the ramp.

If you change either of these, you will change the shape of the curved path the ball takes. Try this – allow your students to play with the apparatus. Put a mark on the bottom edge of the ramp and see if they can adjust the path of the ball to hit it!

Teaching and Pedagogy

Both Galileo and Newton used ramps to study the effects of falling bodies, but Newton brought a much more mathematically sophisticated approach to the matter, employing algebra, graphing, and even the calculus which he developed entirely by himself. Ramps are useful in studying gravity because they allow us to slow everything down and more easily see what is happening.

In our activity, we use the horizontal marks on our graph to represent time. This works for us because the ramp is tilted in only one direction (the vertical); it is perfectly flat in the other direction (the horizontal). Because our ramp isn’t tilted from left to right, the ball’s horizontal speed is unchanged as it rolls across our ramp; the ball takes the same amount of time to cross each square of the graph from left to right!

Younger children may find this hard to grasp, but it is easy to demonstrate to them (and to yourself!) Place the ramp flat on the table and place a marble on it. Now lift any side of the ramp you wish – which way does the ball run? The ball runs downhill each time of course. Ask your students why the ball doesn’t run sideways when you raise one side of the ramp? The children will be quick to tell you that the ramp doesn’t tilt that way – exactly correct!

The curved line lets us see quite easily that the ball speeds up only in the downhill direction – and this is the acceleration of gravity at work! Newton went quite a bit farther of course, he related the acceleration in the fall of an apple to the acceleration causing the orbital path of the Moon to curve around the Earth instead of flying off into space!

Student Outcomes

What will the student discover?

  1. Gravity is a steady force that never changes. We can manage gravity’s effects with ramps and pendulums, but we can never change gravity itself.
  2. Gravitational acceleration is the steady increase in speed as gravity pulls on a falling object. Gravity accelerates things whether they fall freely toward the ground, or roll freely down a ramp or hill.
  3. Horizontal motion cannot effect gravity. Things fall or roll down ramps at the same rate no matter how fast or slow they move horizontally. Vertical and horizontal motion are independent.

What will your students learn about science?

  1. Once again we see that a simple model can reveal wonderful secrets of nature. Gravity has been studied since ancient times, and yet the Packard Apparatus was not developed until a high school teacher invented it in 1906.
  2. Wonderful mathematics lurk in the most surprising places. The curve traced out by the rolling ball is a parabolic curve. You may remember studying parabolas when you took algebra in high school, but you were likely not exposed to these curves in this simple and natural manner.

Conducting the Activity

Materials

  1. A hard, flat surface for a ramp, at least 24 inches square. A piece of Masonite works well for this, but even a tilted school desk will do in a pinch!
  2. A ruler with a groove down the middle. (Alternatively, you can use a 6-inch long piece of wooden corner protector.
  3. A large (25mm) marble or ball bearing. The marble should be relatively heavy, glass or metal balls work well, wooden or plastic balls will not.
  4. A foam stamp pad, well filled with ink. A thin kitchen sponge saturated with paint in a disposable food container can be substituted for this – but ink works better.
  5. 1 piece of very fine (200 grit or higher) sand paper (a common emery board works well for this!)
  6. A pair of rubber kitchen gloves or similar
  7. Construction paper, rulers, markers, masking tape

Building the Packard Gravity Ramp

  1. [Teacher] The marble or ball bearing must be sanded a bit to help it hold ink well. Take your emery board or sand paper and rub the entire surface of the marble vigorously while wearing your kitchen gloves. Do this over a sink or pan of water – the dust from sanding glass can be very abrasive! Rinse the marble occasionally as you sand, when the entire surface has lost its polish and is uniformly dull, you are done. Rinse and dry the marble completely.image
  2. Use a ruler and draw a grid of ½-inch (1 cm) squares over the entire construction paper. Draw as neatly as you can for best results. If you have large format graph paper, feel free to use it here!
  3. Tape your construction paper graph to the ramp board and then prop one end up 4-5 inches with text books so that it is sturdy. You may wish to cover your table with newspaper before you do this activity – the inky marble can be a bit messy!

Exploring the Packard Gravity Ramp

  1. Use the ruler with the groove in it as a marble launcher. Allow the marble to roll down the launcher as shown. Practice a few times with a dry marble; ideally, the marble should start at the upper left corner of the ramp and roll off the bottom right corner – this will give you the best results.
  2. Once you have this down, put on a kitchen glove and rub the marble over the stamp pad until it is thoroughly covered in ink. Carefully set the ink-covered marble on the launcher and let it go. It should trace a neat, curved path across your graph paper. Allow the ink to dry completely before taking the graph paper off of the ramp. If the line is faint, use a marker or crayon to neatly trace over it to make it more visible.image
  3. Notice that the curve becomes steeper as it moves from left to right across the graph. The steepness of the curve is an indication of velocity. As the ball rolls faster down the ramp, pulled ever faster by gravity, the curve becomes steeper.
  4. The curve is never flat. This tells us that gravity is relentless, speeding the ball ever faster down the ramp as it moves. This continuous speeding up is gravitational acceleration.
  5. The curve is also very smooth and regular. This tells us that gravity’s pull is steady and unchanging. If gravity were changing, we would see wobbles and irregularities in the curve. Because the curve is steady, we know that gravity is, too!

Discussion Questions

  1. What does acceleration mean to you?
    • Answer: Acceleration is the steady increase in speed any object experiences as it falls. We won’t worry about the mathematical description of acceleration here, it is enough for students to know about the increase in velocity due to gravity as something falls.
  2. How did you detect acceleration from your results?
    • Answer: The ball travels farther (in the vertical direction) for every unit of time. To go farther in the same time – you must be moving faster!
  3. What would happen if you changed the angle of the big ramp?
    • Answer: Steeper angles give you higher acceleration. Shallower angles give you lower acceleration.
  4. What would happen if you changed the angle of the little ramp at the top, but kept the big ramp unchanged?
    • Answer: The speed of the ball across the ramp would change – the acceleration down the ramp would not!

 

Supplemental Materials

Going Deeper

imageLike the ball rolling down a ramp, satellites are continuously falling around the Earth. Isaac Newton was the first to realize this.

Newton visualized a ball shot from a cannon on a hill at ever greater speeds. Newton realized that not only would the ball travel farther as the horizontal speed increased, he also realized that if the ball were moving fast enough, it would circle the entire Earth.

This was the first conception of a man-made satellite. Newton’s conception of a satellite launched by man was not realized for 300 years, yet in 1958 the Soviet Union launched Sputnik, the world’s first ‘artificial moon’.

The great speed required for a satellite to reach orbit is over 17,000 mph – that is more than 22 times the speed of sound!

Being an Astronomer

The escape speed for the Moon is much lower than that of the Earth – this too is due to gravity. The Moon has only 1% of Earth’s mass (Earth is 100x heavier). This low mass makes the Moon’s gravity about 1/6th that of the Earth’s, just over 5,300 mph.

imageWhen you compare the escape speed of the Earth, over 25,000 mph, to that of the Moon, it becomes clear that it is easier to launch a rocket away from the Moon than it is to launch a rocket away from the Earth.

Being a Scientist

  1. Have students examine the graph they have made. Each line across the graph from left to right represents one tick of time. The vertical distance the ball has rolled down the ramp starts with zero at the top and is numbered down the page.
  2. Look at how far down the ramp the ball rolls for each tick of time. Have the students draw triangles as shown below to help them see this idea in action. What do they notice?
  3. If everything has worked properly, the students should notice that the vertical leg of the triangle gets larger each time because the ball continues to speed up as it rolls down the ramp. Congratulations, you’ve discovered acceleration!

Following Up

There are many excellent videos and documentaries about the journey from the Earth to the Moon. Find one of these and watch it with your class. The sense of history is as important in science as the sense of the future.

Activity 18. Einstein’s Gravity: The Curvature of Spacetime

The curvature of spacetime and Einstein’s concept of gravity may seem like quite a stretch for a non-college classroom; allow me to assure you that it is not! Newton’s theory of gravity is no less complex and subtle, yet we feel comfortable with it through long association. Einstein’s gravitational model is also powerful and mathematically subtle, but like Newton’s ideas of gravity, we can demonstrate it simply with a classroom model that students can grasp cognitively without troubling them (or you!) with the higher mathematics of the subject.

Though you may not realize it, Newton’s theory has a huge hole in it. Newton tells us that orbits work because of the force of gravity – a force that pulls all things together. So far so good, but Newton completely sidesteps the issue of how gravity works, he simply insists that it does work, and gives us convincing mathematical models that show us what will happen when any object is in orbit around another. Einstein stepped in and filled in that gap in Newton’s theory some 350 years after Newton first published his ideas on gravitation.

Einstein’s idea was a simple one, although it may seem a bit odd to you at first. Einstein said that space and time were not separate, but in fact one thing which he called spacetime. Einstein said that spacetime was a fabric which could be bent and stretched by massive objects. Things like stars and planets caused spacetime to curve, and it was this curvature which we call gravity. If this all seems strange, do not worry – you (and your students!) will see how it works as you build and work with this new model of Einsteinian spacetime-gravity.

Academic Standards

Science and Engineering Practices

  • Developing and using models.
  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Argument from evidence.

Crosscutting Concepts

  • Cause and effect.
  • Systems and system models.

Next Generation Science Standards

  • Forces and interactions (K-5, 6-8, 9-12).
  • Engineering and design (K-5, 6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. Einstein was a visual thinker. The models we are making reflect Einstein’s creative, visual, conceptual thinking.
  2. Einstein saw space and time not as separate things, but as one united thing he called spacetime.
  3. The fabric of spacetime can bend, stretch, and curve. Einstein saw gravity as the curvature of spacetime.

Teaching and Pedagogy

This model is a wonderful toy; students and adults alike seem to find playing with it irresistible. Like all the best models, this simple device shows us quickly, and intuitively how the universe around us works. Draw your attention back to Galileo and Aristotle. Galileo challenged Aristotle’s ideas by doing experiments that highlighted the weaknesses in Aristotle’s theories. Scientists of the 17th century were forced to abandon Aristotle’s theories in favor of those of Copernicus and the sun-centered solar system. Einstein did much the same thing in the early 20th century by challenging Newton’s theory of gravitation.

Much like Galileo, Einstein found that challenging an old established figure and a cherished scientific theory did not make him popular. However, when a critical experiment in 1919 proved that stars do bend spacetime – and the light that shines past them – Einstein became famous almost overnight.

Perhaps more than any other model we discuss in this book, Einstein’s model of gravitation needs to be played with. Rolling BB’s or small ball bearings across the empty fabric, then adding a massive marble or weight and trying it again. Actually seeing gravity in action, not as a mysterious force, but as the common sense action of a visual model is uniquely powerful for all students.

Student Outcomes

What will the student discover?

  1. Gravity is not a mysterious attracting force. Rather it is the simple and logical reaction to a mass moving over a curved surface.
  2. It is relatively simple to understand how gravity works – much easier than learning how to do the math involved!
  3. Masses (like planets and stars) tell spacetime how to curve.
  4. The curvature of spacetime tells mass how to move.

What will your students learn about science?

  1. Powerful new ideas are often visual or physical in nature – the mathematics often comes later, sometimes many years later!
  2. Our understanding of complex ideas often first comes to us visually, often viscerally, long before we can describe or explain what we know.
  3. To bridge the gap between our first understanding of an idea, and our ability to explain or discuss what we know, a great deal of play and experimentation is often essential.

Conducting the Activity

Materials

  1. A 30-inch plastic hula-hoop (or similar)
  2. Eight, 15-inch pieces of 1-inch PVC pipe
  3. Eight, 1-inch PVC T-connectors
  4. Small can of PVC cement or super glue
  5. 1 yard of black spandex fabric or similar (must be stretchy in both directions!)
  6. Two billiard balls
  7. Several glass marbles of various sizes, BB’s or small ball bearings may also be used

Building Einstein’s Gravity Model

  1. Your local home improvement store almost certainly does not have 15-inch sections of PVC piping, you will have to buy a longer piece and cut these yourself. You can easily cut PVC piping with a hacksaw, or with a PVC pipe cutter. Once you have these pieces cut, glue them securely into the T-connectors so that they each form a rigid T-shape.
  2. Once the glue is completely dry, measure exactly 12-inches down from the T-connector and cut the PVC pipe off there – this will insure that all the pieces are the same length. These will be the legs for your model to stand on and they must be the same length for everything to be level and work correctly.
  3. The T-connectors now need to be cut down as shown below so that they will snap around the hula-hoop. This can be done with a hacksaw, but I find a belt sander to be easier and faster. Check with your custodial department for help with this!
  4. Cut a circle of spandex fabric so that it is about 6-inches larger than your hula hoop. Stretch the fabric over the hoop and snap one of the T-connector legs over the fabric to hold it in place. Work on opposite sides of the hoop, stretching the fabric and snapping in the connectors until the fabric is stretched tightly over the entire hoop. You should now have a 30-inch ‘trampoline’ of spandex fabric on short legs of PVC piping. Your model is now ready to use.

Exploring Einstein’s Gravity Model

  1. Explain to your students that the black fabric represents the fabric of spacetime. Roll one of your small glass marbles across the fabric and observe what happens – it will roll straight across the circle.
  2. Now place a billiard ball by itself in the center of the fabric – what happens? The fabric is stretched! Mass tells spacetime how to curve! Try different size marbles and weights, let the students see that greater mass causes greater curvature of the spacetime fabric.
  3. With the billiard ball resting in the center of the fabric, try rolling a small marble straight past the billiard ball. It will not roll straight – it curves toward the larger ball. The curvature of spacetime tells mass how to move!

Ask your students why the marble curved this time when it rolled straight before? They will quickly realize that the larger ball has stretched and curved the black fabric – it is the curved shape of the fabric that causes the marble’s path to bend toward the larger ball.

This is how gravity works! This is Einstein’s explanation for gravity. A large mass causes the spacetime fabric to curve – and the curved shape of spacetime controls how the masses have to move.

4. Try placing two billiard balls several inches apart from each other on the fabric and observe what happens. The curvature of the fabric causes them to roll toward each other – just as gravity causes all things to be pulled together.

5. Can your students get some of the smaller marbles to orbit around one of the billiard balls? Can they get two billiard balls to orbit around each other? Have fun and play with this model for a while – it is fascinating to everyone who sees it and students will easily see how Einstein’s spacetime fabric creates the effect we call gravity.

Discussion Questions

  1. How does Einstein’s gravity work?
    • Answer: “Mass tells spacetime how to curve. The curvature of spacetime tells mass how to move.” – John Wheeler. This elegant quote from one of Einstein’s greatest students explains things perfectly!
  2. How is Einstein’s model of gravity better than Newton’s model?
    • Answer: This is likely to generate a lot of comment and discussion, but essentially Einstein’s model explains how gravity works, Newton’s model simply tells us what gravity does, but fails to explain how it works.
  3. Robert Hooke’s pendulum model showed how planets orbit a star, can you make a model of an orbiting planet or moon with this model?
    • Answer: Yes! The large billiard ball acts nicely as a star while the small marble acts as a planet in orbit. Like Hooke’s model, you will find it almost impossible to create a circular orbit with Einstein’s model – and for the same reason. Creating the perfect balance between gravity and momentum is hard and friction will cause any circular orbit to quickly decay into an ellipse in any case.

Supplemental Materials

Going Deeper

Sadly, when we add mathematics to a science lesson, we often end up teaching math instead of science. Let’s do something different and use the science concepts to help understand what the math means.

One of the first lessons students learn in physics is sometimes called the free fall equation; this simple equation multiplies two numbers to find out how far something falls in a given amount of time. The equation looks like this:

h = g * t

In this equation, h stands for height (the distance an object falls); g stands for the acceleration of gravity; and t stands for the falling time in seconds. In other words, if you know the value of g (9.81 m/s2) and the number of seconds an object is falling, you can calculate how far it falls.

Unfortunately, we tend to fall back on something like: “Multiply the two numbers and get the answer for how far the rock falls.” Let’s see if we can help even our youngest students understand what the math means.

t for time is pretty simple, this is how long anything spends falling. It doesn’t matter if it is a rock falling off a cliff or a swimmer falling off a diving board. So what does g mean? g is the curvature of spacetime! Place a billiard ball in the center of our model to represent the Earth and note how much the fabric curves. Allow a BB to roll toward the billiard ball – see how fast it rolls?

Replace the billiard ball with a large marble – the fabric curves much less now! Let this marble represent our Moon, now allow a BB to roll toward the marble – see how much more slowly it rolls? The gravity on the Moon is less (we fall more slowly) because the curvature of spacetime is less. The gravity on the Earth is more (we fall faster than on the Moon) because the curvature of spacetime is greater here.

And that little g in our equation? g represents the curvature of spacetime. Now we understand the math much more completely. Multiply the falling time by the curvature of spacetime, and you find out how far an object falls. The simple equation is no longer just a multiplication problem, we now understand what each number means, and how gravity works!

Being an Astronomer

Let’s look at our gravity model again. Place a billiard ball in the center of the model and look closely at the fabric. Do you notice how the fabric has the greatest curvature right under the billiard ball, but the fabric becomes more flat (less curved) as you move away from the billiard ball.

If gravity works because of the curvature of the fabric of spacetime, then our model seems to suggest that gravity gets weaker as you move away from a planet or star. What does this mean for things in orbit like spacecraft or moons circling a planet?

Start with a marble and see if you can get it to orbit around our billiard ball planet. Do you notice how the marble moon eventually spirals into the billiard ball planet? What do you notice about the marble’s speed as it spirals in? That’s right! The marble moon moves faster as it gets closer to the planet it orbits. Let’s see if we can confirm this prediction made by our marble with our own observations. Do things that are farther away from the Earth move more slowly across the sky?

Find a clear night and begin observing about 30 minutes after sunset. In a dark sky, you will be able to spot satellites moving across the sky. These satellites look like small stars that drift noticeably across the fixed stars in the constellations. You will find that these satellites move fast enough for you to easily detect their motion. If you could time them all the way across the entire sky, they would complete their journey in a matter of minutes.

Now consider the Moon. As we have seen in previous activities, you can track the motion of the Moon across the sky from east to west, but this takes about 14 days. The Moon is also much farther away than any man-made satellite. Our Moon orbits at a distance of about 385,000 km while artificial satellites that we can see orbit at a distance of 100-250 km from the Earth.

Our observations back up what our model tells us about gravity. The force of gravity is stronger near the Earth than it is far out in space; and satellites that orbit closer to the Earth travel faster – just as the marbles do when they orbit around a billiard ball in our gravitational model!

Being a Scientist

One hundred years after Einstein finished his work on relativity theory, scientists are still working to design and build experiments to confirm the predictions of Einstein’s theories today. One of these experiments is called LIGO, and it is used to detect gravitational waves.

Einstein predicted that if spacetime was indeed a unified fabric, that there should be waves in spacetime just as there are waves in a pond when you throw a stone into the water. Investigate LIGO on the internet – what would anyone do with a ‘telescope’ that detects gravitational waves?

Following Up

Einstein is famous for being the first man in 250 years to correct or adjust Newton’s theory of gravitation. The idea of becoming famous for correcting someone else’s work is common in the history of science. What other scientists and astronomers can you find that have become well known for making corrections or improvements in someone’s earlier work?

On the other hand, what astronomers or scientists can you find that are famous for doing their own original work? I’ll give you a hint… there are some from each group in this book!

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Astronomy for Educators by Daniel E. Barth, PhD is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book