History of Gravity. Name: Date: Period:

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1 History of Gravity Name: Date: Period: I. ANCIENT ASTRONOMY Imagine what it was like for the first humans to look up at the night sky. This is well before the invention of modern technology. There were no telescopes, no cameras, and certainly no computers. Try to imagine what it was like for ancient humans to look up at the night sky. (a) As time passes during the night, what do we see happen to the stars in the sky? (b) How do you think ancient humans would have explained this motion? 2. THE WANDERERS As time passed, people started making closer and closer observations of the stars. Instead of just looking at the sky over the course of a single night, astronomers started to look at the stars over a series of nights. They noticed that not all of the stars moved together. In fact, five of them appeared to wander across the sky. They called these the wanderers or in Greek: planets. (a) How did the ancient Greek know that the planets were different from the stars? (b) Why, do you think, did the ancient Greeks call the planets wanderers? 3. ARISTOTLE S MODEL Aristotle applied his formal system of logic to explain the motion of the planets and stars. He proposed that the planets and stars are each attached to giant crystalline spheres which all spin around Earth. (a) Why did the spheres which carried the planets have to be made of crystal? (b) Why do you think Aristotle said the solar system was geocentric? 4. NEW EVIDENCE As time passed, new tools were invented which allowed astronomers to keep detailed records of the positions of the planets in the night sky. They could now track the location of the planets not just over several days, but over years! They found that the planets do not move in circles like Aristotle thought. (a) Along what path do the planets move across the sky? Describe it! Fremont Physics Kepple 2012 (b) What does this new evidence tell us about Aristotle s model of the solar system? Gravitation Page #1 10/23/12

2 5. PTOLEMY S MODEL Ptolemy attempted to fix Aristotle s model by saying that planets were attached to a smaller crystal sphere which was itself attached to the larger crystal spheres. These smaller spheres would move around in what he called an epicycle while the larger sphere moved in a circle around the Earth. (a) Draw a picture of the path Mars follows around the Earth according to Ptolemy. (b) Summarize how Ptolemy explained the retrograde motion of Mars. 6. A MORE SIMPLE EXPLANATION About 1,500 years after Ptolemy, an astronomer named Copernicus revolutionized our understanding of the solar system. Copernicus said that the retrograde motion of Mars could be more easily explained if we simply placed the Sun at the center of the solar system and set the Earth and all the other planets in circular motion around the Sun. (a) Summarize how Copernicus explained the retrograde motion of Mars. (b) Contrast the main differences between Ptolemy s and Copernicus s models. 7. THE LAW OF HARMONY Using the best astronomical data available at the time, Kepler created a precise mathematical model of the solar system based entirely upon detailed observation data of the planets. His model proved the heliocentric model proposed by Copernicus, with one exception: the planets actually orbit on elliptical (oval) paths, not perfect circles. In addition, Kepler also discovered that the cube of the radius of a planet s orbit is equal to the square of its period. This relationship is known as the law of harmony. Our long journey towards understanding the solar system has finally led to a mathematical model which can, for the first time, accurately predict the position of the planets! (a) Summarize the relationship that Kepler discovered between all the planets. (b) Write down the mathematical expression for the law of harmony:

3 Universal Gravitation Name: Date: Period: I. BE NEWTON FOR A DAY When Isaac Newton was 22 years old, he derived the law of universal gravitation. This is how he did it 1. Along what path does a body travel when there is no force acting on it? 2. Along what path does the Moon travel? 3. What is required to travel in a circle? 4. What does this say about the Moon? 5. What relationship did Newton discover between force, mass, and acceleration? 6. What is the equation for the acceleration of a body in circular motion? 7. What is the speed in circular motion? 8. Square all the terms in part 7 to find v². 9. Substitute the speed-squared into part Substitute the acceleration into part 5. In our last lesson we learned about Kepler s law of harmony: the radius of a planet s orbit cubed divided by the square of the planet s period is a constant value Rearrange Kepler s Rule to solve for 𝑇². 12. Substitute 𝑇² into part 10. =𝐾 𝑇2 In an effort to explain the law of harmony, Newton created his hypothesis of universal gravitation: any two masses in the universe will attract each other. The force acting on the moon must come from the mass of the earth pulling on the mass of the moon. Since our expression for this force in part 12 only has the mass of the moon, the mass of the earth must be hidden inside Kepler s constant! 13. Start with the constant pieces from part 12, split off M from K and name what s leftover G. 14. Substitute the result from 13 into part 12. 4𝜋 2 𝐾 = 𝐺𝑀 Fremont Physics Kepple 2012 Gravitation Page #2 10/27/14

4 II. UNIVERSAL GRAVITATION According to Newton s universal gravitation, the strength of the gravitational attraction between two objects depends only on the mass of each object and the distance between the two objects. Newton said it like this: gravity varies directly with mass and inversely with the square of the distance. 15. Write the law of universal gravitation. 16. What does the 𝐺 stand for? 17. Is there another way to write the law of universal gravitation without using 𝐺? 18. Write universal gravitation as a proportionality in order to remove 𝐺. 19. According to universal gravitation, what is the relationship between the force of gravity and (a) mass? (b) distance? III. GRAVITY PROPORTIONS Two spheres gravitationally attract one another. The force between them is measured to be 100 N. Draw an arrow on the diagram to represent the force acting on each mass. 20. Calculate the gravitational force if the mass of the left sphere were doubled. Draw an arrow on the diagram to represent this new force Calculate the gravitational force if the distance between the two masses were doubled. Draw an arrow on the diagram to represent this new force.

5 Gravity Proportions Name: Date: Period: I. INVERSE SQUARE LAW Newton said the strength of gravity varies inversely with the square of distance. This is known as the inverse square law As the distance increases, the influence of gravity is spread out over a greater area, which reduces its overall strength. 1. Use the inverse square law to determine the missing factor. (a) 2 (b) 1 2 (c) 3 (d) 1 5 (e) 1 4 (f) 9 (k) 16 (l) II. WHAT IF BOTH DISTANCE AND MASS CHANGE? Two spheres gravitationally attract one another. The mass and distance of the spheres are unknown, but we measure the force between them to be 20 N. For each situation, draw a new picture and use universal gravitation to calculate the new force. 2. right mass: /4 left mass: 2 distance: / N 3. right mass: 3 left mass: 2 distance: /2 4. right mass: /2 left mass: 3 distance: 2 Fremont Physics Kepple 2012 Gravitation Page #3 10/27/14

6 III. PROPORTION PROBLEMS The force of gravity between two objects of mass separated by distance 𝑟 is measured to be 100 N (pictured to the right). Using this information, answer each of the following questions for the arrangement of masses shown below where each mass is separated by the same distance. 100 N A B C D E F 4 3 / What is the force between masses A and E? 6. What is the force between masses C and E? 7. What is the force between masses A and B? 8. What is the force between masses A and C? 9. What is the force between masses D and F? 10. What is the force between masses B and E? 11. What is the force between masses C and D? 12. What is the force between masses C and F?

7 Acceleration of Gravity Name: Date: Period: I. WHAT EXACTLY IS LITTLE G? While standing on a planet, the mass of the planet pulls the mass of your body downward. This force is called your weight; it is a measure of the gravitational attraction between your body and the Earth. 1. Applying Newton s 2nd Law, what is the formula for weight? 2. Applying universal gravitation, what is the formula for weight? 3. At the surface of Earth, what two masses and one distance determine your weight? 4. Combine Newton s 2nd law and universal gravitation to find an expression for. 5. The radius of Earth is meters and the mass of Earth is acceleration of gravity at the surface of Earth? kilograms. What is the II. NEWTON TESTS HIS HYPOTHESIS Newton s hypothesis of gravity was that any two masses attract each other. In order for his hypothesis to advance to the status of scientific theory, it would have to be tested. Newton s test was to calculate the distance the moon falls in one second. He would make this calculation first using geometry and then using universal gravitation. If the two different methods gave the same result, then he would know that his hypothesis was correct. 6. Using geometry, Newton calculated that the moon falls 1.4 mm in one second. Now we ll use universal gravitation to calculate this distance. (a) What is the acceleration of gravity from Earth at the moon s orbit? (b) How far does the moon fall in one second? 7. What does this result tell us about Newton s hypothesis of gravity? 8. Now that the hypothesis has been tested, what do we call it? Fremont Physics Kepple 2012 Gravitation Page #5 10/28/13

8 9. While standing on Planet, with radius of and a mass of, you measure the acceleration of gravity to be 10 m/s². The table below gives the radius and mass of several other planets. Your job is to determine on each of the other planets. Planet Planet Planet Planet Planet,,,,, (a) What is at the surface of planet? (b) What is at the surface of planet? (c) What is at the surface of planet? (d) What is at the surface of planet? 10. On which planet would you weight the most? Planet Planet Planet Planet Justify your answer: Planet 11. Your friend thinks you would weigh the most on Planet. She says that the force of gravity gets stronger with more mass; therefore you would weigh more on the biggest planet. Explain why your friend is wrong.

9 Name: Gravitation Practice Test Date: Period: I. VOCABULARY If the underlined word is used incorrectly, cross it out and replace it with the correct word. 1. The law of universal acceleration says: there is an repulsive force between any two masses. 2. Newton found that the force of gravity varies directly with the distance and inversely with the square of the force. 3. When the distance between two objects is tripled, the force is one-fourth as strong. 4. Gravity is a very strong force, it is barely noticeable unless the masses are small. 5. Your weight on a planet depends on the shape and color of the planet. II. FORCE OF GRAVITY DRAWINGS 6. Draw arrows to indicate the magnitude of the force of gravity acting between each pair of masses. III. MULTIPLE CHOICE 7. Which of the following best explains why an object has less weight on top of a mountain? A B C D Because you are closer to space. The atmosphere is lighter. The mountain adds more mass. The mountain adds more distance. 8. On a planet with twice the radius and twice the mass of the earth, your weight would be equal to your weight on earth times a factor of A 1/4 B 2 C 1/2 D 4 IV. GRAVITY PROPORTION PROBLEMS 20 N 9. What is the gravitational force on mass A from (a) mass B? (show work!) (b) mass C? (show work!) 20 N??𝑁??𝑁 B A C 2 4 Fremont Physics Kepple 2011 Gravitation Page #6 10. What is the net force on mass A? 11. In which direction will mass A accelerate?

10 Gravitation Study Guide MAJOR CONCEPTS: The law of universal gravitation states that any two masses will attract each other. The force of gravity is directly proportional to mass and inversely proportional to the square of distance. The force of gravity extends infinitely in all directions, but it gets weaker with distance. FORCE OF GRAVITY DRAWINGS The force of gravity increases with mass and decreases with distance. The length of the arrow indicates the strength of the force. The longer the arrow, the stronger the force. Gravity is a very weak force, it is barely noticeable unless the mass involved is huge. Your weight on a planet depends only upon the mass and radius of the planet, not the size of the planet. Because the moon has tangential velocity, it orbits Earth instead of crashing into it. FORCE OF GRAVITY PROPORTIONS The relations within the law of gravitation are FORCE and MASS are DIRECT! STRONGER (more mass) FORCE and DISTANCE are INVERSE SQUARE! 2 m F 2 r F WEAKER (more distance) 2 m F 2 r F GRAVITATION PROPORTION PROBLEMS If the force of gravity between two masses is given, you can figure out the other forces. B 20 N?? N A 20 N m m m?? N 2m m 3m C Ex1. What is the force between masses A and B? F Mm r 2 2 F AB 2F Ex2. What is the force between masses B and C? F Mm 2 3 r F BC 3 2 F

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