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GENERAL PHYSICS 1 Quarter 2 – Module 2 GRAVITY Department of Education ● Republic of the Philippines Senior High School

GENERAL PHYSICS 1

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Quarter 2 - Module 2: GRAVITY

First Edition, 2020

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Table of Contents How to learn from this Module.................................................................................................................i Icons of this Module....................................................................................................................................i What I Know................................................................................................................................................. ii Lesson 1: NEWTON’S LAW OF UNIVERSAL GRAVITATION What I Need to Know...................................................................................................... 1 What’s In ………………………………………………………………………… What’s New: ..................................................................................................................... 2 What Is It............................................................................................................................. 3 What’s More: (Activities) .............................................................................................. 4 Lesson 2: GRAVITATIONAL FIELD What’s In............................................................................................................................. 4 What I Need to Know...................................................................................................... 5 What Is It ........................................................................................................................... What’s More .................................................................................................................... Lesson 3: GRAVITATIONAL POTENTIAL ENERGY What’s In............................................................................................................................. 6 What I Need to Know...................................................................................................... 6 What Is It (Discussion)................................................................................................... What’s More..................................................................................................................... Lesson 4: ORBITS What’s In............................................................................................................................. 8 What I Need to Know...................................................................................................... 8 What Is It ........................................................................................................................... What’s More..................................................................................................................... Lesson 5: KEPLER’S LAWS OF PLANETARY MOTION What’s In............................................................................................................................. 11 What I Need to Know...................................................................................................... 11 What Is It (Discussion)................................................................................................... What’s More:.................................................................................................................... Assessment: (Post-Test) ……………………………………………………………………….. Key to Answers........................................................................................................................................... 15 References................................................................................................................................................... 16

What I Know Multiple Choice. Select the letter of the best answer from among the given choices.

1. The gravitational constant has a symbol of __________. A. C C. k B. G D. E
2. Who is the proponent of the Universal Law of Gravitation? A. Newton C. Archimedes B. Einstein D. Pascal
3. The symbol Fg is referred as the magnitude of the ___________. A. Electrical Force C. Gravitational Force B. Rotational Force D. Magnetic Force
4. What is the value of G (in three significant figures)? A. 6.36 x 10-8^ N C. 6.67 x 10-11^ N m^2 / kg B. 9.11 x 10-36^ kg D. 9.8 m / s^2
5. In which of the following cases would you feel weightless? A. Traveling through space in an accelerating rocket B. Falling from an airplane with your parachute open C. Walking on the moon D. Taking an exam
6. A useful way to describe forces that act a distance is in terms of __________. A. Newtons C. distance B. Period D. field
7. If your mass is 65 kg on Earth, what would your mass be on the Moon? A. 11 kg B. 11 lb C. 65 kg D. 65 N
8. How much GPE does a 2-kg block have if it is lifted 12.5 m high? A. 480 J B. 490 J C. 500 J D. 510 J
9. Which of the following is NOT one of Kepler’s Laws of Planetary Motion? A. The square of a planet’s period is proportional to its distance from the sun cubed. B. The area of a planet’s orbital plane is inversely proportional to its speed. C. A planet sweeps out equal area in an equal time interval. D. Planets move around the sun in elliptical orbits.
10. He was a Danish astronomer who worked with Kepler. His data collected were used by Kepler in his laws. A. Aristotle C. Tycho Brahe B. Archimedes D. None of them

ii

Lesson NEWTON’S LAW OF 1 UNIVERSAL GRAVITATION Some of the earliest investigations in Physical Science started with questions that people asked about the night sky. Why doesn’t the moon fall to the earth? Why do the planets move across the sky? Why doesn’t the earth fly off into space rather than remaining in orbit around the sun? The study of gravitation provides the answers to these and many related questions. Gravitation is one of the four classes of interactions found in nature, and it was the earliest of the four to be studied extensively. Newton discovered in the 17th^ century that the same interaction that makes an apple fall out of a tree also keeps the planets in their orbits around the sun. In this module, you will learn the basic law that governs gravitational interactions. This law is universal : Gravity acts in the same fundamental way between the earth and your body, between the sun and the planet, and between a planet and one of its moons. We’ll apply the law of gravitation to phenomena such as the variation of weight with altitude, the orbits of satellites around the earth, and the orbits of planets around the sun. What I Need to Know At the end of this module, you should be able to:  Use Newton’s Law of gravitation to infer gravitational force, weight, and acceleration due to gravity (STEM_GP12G-IIb-16); What’s In The example of gravitational attraction that’s probably most familiar to you is your weight , the force that attracts you toward the earth. During his study of the motions of the planets and of the moon, Newton discovered the fundamental character of the gravitational attraction between two any bodies. Along with his three laws of motion, Newton published the Law of Gravitation in 1687. It may be stated as follows:

Every particle of matter in the universe attracts every other particle with a

force that is directly proportional to the product of the masses of the particles

and inversely proportional to the square of the distance between them.

What Is It Calculating Gravitational Force Example: The mass m 1 of one of the small spheres of a Cavendish balance is 0.0100 kg, the mass m 2 of the nearest large sphere is 0.500 kg, and the center-to-center distance between them is 0.0500 m. Find the gravitational force Fg on each sphere due to the other. Solution: Because the spheres are spherically symmetric, we can calculate Fg by treating them as particles separated by 0.0500 m. Each sphere experiences the same magnitude of force from the other sphere. We use Newton’s law of gravitation to determine Fg : Acceleration due to Gravitational Attraction Example: Suppose the two spheres in the previous example are placed with their centers 0.0500 m apart at a point in space far removed from all other bodies. What is the magnitude of the acceleration of each, relative to an inertial system? Solution: Each sphere exerts on the other a gravitational force of the same magnitude Fg , which we found in the previous example. We can neglect any other forces. The acceleration magnitudes a 1 and a 2 are different because the masses are different. To determine these, we’ll use Newton’s second law:

What’s More PROBLEM SOLVING ACTIVITY

1. The moon has a mass of 7.34  10 22 kg and a radius of 1.74  106 meters. If you have a mass of 66 kg, how strong is the force between you and the moon?
2. A distance of 0.002 m separates two objects of equal mass. If the gravitational force between them is 0.0104 N, find the mass of each object.

Our solar system is part of a spiral galaxy like the figure below, which contains roughly 10^11 stars as well as gas, dust, and other matter. The entire assemblage is held together by the mutual gravitational attraction of all the matter in the galaxy. What’s More

Answer as required.

1. Compare the gravitational attraction between objects on earth and interaction

of celestial bodies in space. Which gravitational force is almost negligible?

Why?

______________________________________________________________

______________________________________________________________

______________________________________________________________

2. Discuss why the study of a gravitational field is important.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Lesson GRAVITATIONAL POTENTIAL 3 ENERGY What’s In When we first introduced gravitational potential energy in our previous lessons, we assumed that the gravitational force on a body is constant in magnitude and direction. This led to the expression U = mgh. But the earth’s gravitational force on a body of mass m at any point outside the earth is given more generally by Fg = (GmEm) / r^2 , where mE is the mass of the earth and r is the distance of the body from the earth’s center. What I Need to Know

At the end of this lesson, you should be able to:  Apply the concept of gravitational potential energy in physics problems ; (STEM_GP12RedIIb-19) What Is It Consider a block with mass, and is tied to the end of a rope and goes up over a pulley while the other end is being pulled by a man. If the man lets go of the rope, the rope will be pulled downward with a force equal to the force of gravity of the block. The work performed by the block depends on the weight and height, Δh. The work done will be: W = Fd Since F = mg, then: W = (mg)( Δh) where d = Δh 6 The more work to perform and energy stored in the block, the higher the block is from the ground. Gravitational Potential Energy, Eg , is the energy stored of an object because of its distance above the surface of the Earth. The change in gravitational potential energy of an object is expressed as: Eg = mgΔh where: m – is the mass of the object in kilograms g – is the acceleration due to gravity at 9.8 m/s^2 Δh – is the vertical displacement of the object in meters ΔEg - is the object’s change in gravitational potential energy in Joules Sample Problems:

1. How much gravitational potential energy does a 4.0 kg block has if it is lifted 25 m? Eg = mgΔh = (4.0 kg) (9.80 N/kg) (25 m) = 9.8 x 10^2 J
2. A 61.2 kg boy fell 0.500 m out of the bed. How much potential energy is lost? Eg = mgΔh = (61.2 kg) (9.80 N/kg) (-0.500 m)

= -299.8 J ≈ 300 J

Lesson ORBITS 4 What’s In Artificial satellites orbiting the earth are a familiar part of modern technology. But how do they stay in orbit, and what determines the properties of their orbits? We can use Newton’s laws and the law of gravitation to provide the answers. We’ll see in the next section that the motion of planets can be analyzed in the same way. What I Need to Know At the end of this lesson, you should be able to:  Calculate quantities related to planetary or satellite motion ; (STEM_GP12RedIIb-20) What Is It A circular orbit, is the simplest case. It is also an important case, since many artificial satellites have nearly circular orbits and the orbits of the planets around the sun are also fairly circular. The only force acting on a satellite in circular orbit around the earth is the earth’s gravitational attraction, which is directed toward the center of the earth and hence toward the center of the orbit. This means that the satellite is in uniform circular motion and 8 its speed is constant. The satellite isn’t falling toward the earth; rather, it’s constantly falling around the earth. In a circular orbit the speed is just right to keep the distance from the satellite to the center of the earth constant. The radius of the orbit is r, measured from the center of the earth; the acceleration of the satellite has magnitude arad = v^2 / r and is always directed toward the center of the circle. By the law of gravitation, the net force (gravitational force) on the satellite of mass m has

magnitude Fg = GmEm / r^2 and is in the same direction as the acceleration. Newton’s second law (F = ma) then tells us that GmEm = mv^2 r^2 r Solving this to find the circular orbit, we find This relationship shows that we can’t choose the orbit radius r and the speed independently; for a given radius r, the speed for a circular orbit is determined. The satellite’s mass m doesn’t appear in the equation above, which shows that the motion of a satellite does not depend on its mass. If we could cut a satellite in half without changing its speed, each half would continue on with the original motion. For example, an astronaut on board a space shuttle is herself a satellite of the earth, held by the earth’s gravitational attraction in the same orbit as the shuttle. The astronaut has the same velocity and acceleration as the shuttle, so nothing is pushing her against the floor or walls of the shuttle. She is in a state of apparent weightlessness, as in a freely falling elevator. True weightlessness would occur only if the astronaut were infinitely far from any other masses, so that the gravitational force on her would be zero. Apparent weightlessness is not just a feature of circular orbits; it occurs whenever gravity is the only force acting on a spacecraft. Hence it occurs for orbits of any shape. We can derive a relationship between the radius r of a circular orbit and the period T , the time for one revolution. The speed is the distance traveled in one revolution, divided by the period: To get an expression for T , we use the equation above to solve for T and substitute: We have talked mostly about earth satellites, but we can apply the same analysis to the circular motion of any body under its gravitational attraction to a stationary body. Other examples include the earth’s moon and the moons of other planets. Sample Problem: You wish to put a 1000-kg satellite into a circular orbit 300 km above the earth’s surface. What speed, period, and radial acceleration will it have? 9 Solution: The radius of the satellite’s orbit is r = 6380 km + 300 km = 6680 km = 6.68 x 10^6 m. The orbital speed is

(d) What is its radial acceleration? Lesson KEPLER’S LAWS OF 5 PLANETARY MOTION

What’s In The name planet comes from a Greek word meaning “wanderer,” and indeed the planets continuously change their positions in the sky relative to the background of stars. One of the great intellectual accomplishments of the 16th and 17th centuries was the threefold realization that the earth is also a planet, that all planets orbit the sun, and that the apparent motions of the planets as seen from the earth can be used to precisely determine their orbits. The first and second of these ideas were published by Nicolaus Copernicus in Poland in 1543. The nature of planetary orbits was deduced between 1601 and 1619 by the German astronomer and mathematician Johannes Kepler, using a voluminous set of precise data on apparent planetary motions compiled by his mentor, the Danish astronomer Tycho Brahe. What I Need to Know At the end of this lesson, you should be able to:  For circular orbits, relate Kepler’s third law of planetary motion to Newton’s law of gravitation and centripetal acceleration ; (STEM_GP12G-IIc-22) 11 What Is It The following are the laws developed by Johannes Kepler:

1. LAW OF ORBITS The first law explains that all planets move in elliptical orbits with the sun at one focus. Kepler’s first law means that planets move around the Sun in elliptical orbits. An ellipse is a shape that resembles a flattened circle. How much the circle is