Models of the Universe (SH1690) PDF

SH1690 THE UNIVERSE AND PHYSICS Constructed in 3000 BCE, the Stonehenge in England was thought to have been an observatory used to predict solar and lunar eclipses. It Th...

SH1690 THE UNIVERSE AND PHYSICS Constructed in 3000 BCE, the Stonehenge in England was thought to have been an observatory used to predict solar and lunar eclipses. It The Earth is always in motion. It spins or gyrates about its axis as it was constructed so that the sun would rise above one of the main stones revolves around the sun. These motions of Earth account for many in the summer solstice. The windows at the cop and sides of the pyramid celestial phenomena that are perceived as natural occurrences. at the Mayan site In Palenque, Mexico, were so arranged that the rooms they lead to are illuminated by the rising sun. It said that during THE MODELS OF THE UNIVERSE AND THE SOLAR SYSTEM equinoxes, the illumination of the sun on the stairs and the base of the stepped pyramid creates the illusion of a crawling serpent, symbolizing The Models of the Universe a god closely related to planet Venus in Mayan mythology. Early humans relied on the skies as their principal means of telling the time, navigation, and knowing when to start planting crops. Some 3 000 years ago, the Egyptians established a 365-day calendar based on Sirius's track. This track also coincided with the annual flooding of the Nile River. The Babylonians and the Assyrians also invented similar calendars to help them determine when to sow and reap crops. Astronomy also influenced architecture. Around 2560 BCE, the Pyramids of Giza in Egypt were constructed so that each side faced north, south, east, or west of a compass to within a tenth of a degree. Also, the three (3) pyramids represent the belt stars of the constellation Orion. Figure 2. The Stonehenge, one of the world’s ancient (and somewhat mystical) wonders Source: https://www.history.com/topics/british-history/stonehenge The Early Universe Humans have come up with several models to understand the Universe. Before the telescope's invention, they had to rely on their senses for a picture of the Universe with much philosophical and religious symbolism. Around 600 BCE, Thales of Miletus proposed that Earth is a disk floating on water. In 520 BCE, Anaximander, also from Miletus, suggested that Earth is a cylinder and that its surface is curved. As civilization flourished, several other models were proposed. These models can be grouped into two categories: geocentric and heliocentric. The geocentric model considers Earth as the center of the Universe. The heliocentric model Figure 1. The Great Pyramids of Giza in Egypt assumes the sun to be the center of the Universe. Source: https://en.wikipedia.org/wiki/Pyramid 06 Handout 1 *Property of STI  [email protected] Page 1 of 11  SH1690 The Geocentric Models backtrack for a while; that is, they move westward before resuming their The following were some geocentric models of the Universe proposed eastward motion. This pattern is called retrograde motion. by the Greeks. In all these models, the Earth and the other heavenly bodies were assumed to be spheres. The Pythagorean Model Pythagoras was acknowledged to be the first to assert that the Earth is round and that the heavenly bodies move in circles. In his model, the Earth is at rest at the center of the Universe, and everything rotates around it. Pythagoras also considered that the motions of the planets were mathematically related to musical sounds and numbers. These ideas are called “The Music of the Spheres.” Figure 4. A simple illustration of how Mars’ retrograde motion occurs as seen on Earth Source: http://va-iitk.vlabs.ac.in/?page=exp3 The followers and students of Plato were called upon to explain the retrograde motion of the planets. In particular, Plato challenged them with this problem: “What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?” Figure 3. An illustration of Pythagoras’ Music of the Spheres This challenge is known in the history of astronomy as “Plato’s saving Source: http://www.sensorystudies.org/picture-gallery/spheres_image/ appearances”. Anaxagoras, a follower of Pythagoras, was credited with having Eudoxus’ Model determined the relative positions of the Sun, the Moon, and the Earth Eudoxus was the first to “save the appearances” that Plato referred to, during solar and lunar eclipses. using a series of 27 concentric spheres on which the Sun, the Moon, and the planets moved in a perfect circular motion. The breakdown of the 27 Plato’s “Saving the Appearances” spheres is as follows: The Greek philosopher and teacher Plato adopted the Pythagorean view One sphere for fixed stars; of the motion of the heavenly bodies as combinations of circular motion about Earth. He assumed that all motions in the Universe are perfectly Three spheres for the sun; circular and that all heavenly bodies are ethereal or perfect. Most of the Three spheres for the moon; and time, planets move from west to east as predicted. But occasionally, they 06 Handout 1 *Property of STI  [email protected] Page 2 of 11  SH1690 Four spheres for each of the five known planets were Mercury, causing the other spheres to rotate as well. According to Aristotle, the Venus, Mars, Jupiter, and Saturn. order of heavenly bodies in the Universe was (from the Earth out): the Earth, The Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, the fixed stars, and the firmament of the Prime Mover. The Roman Catholic Church adopted this idea in Medieval times, where the Prime Mover was considered God and the sphere of the firmament as heaven. Figure 5. The Solar System as viewed and proposed by Eudoxus Source: http://www.astronomy.ohio-state.edu/~pogge/Ast161/Unit3/greek.html Note that he did not assign any sphere for the Earth because the Earth is fixed in his model. The outermost sphere was aligned with the celestial poles rotating once a day to increase and set the effect. The next sphere was tilted 23.5° and rotating slowly to simulate the planet's usual west- to-east movement about the fixed stars. The last two spheres produce the backward motions of the planets. Eudoxus explained the rotation of the 27 celestial spheres using the notion of “intelligences.” Aristotle’s Model Figure 6. The Aristotelian model of the Solar System The Aristotelian model also used the 27 celestial spheres of Eudoxus. In Source: http://homework.uoregon.edu/pub/emj/121/lectures/aristotle.html addition, Aristotle used 27 “buffering” spheres between the celestial spheres of Eudoxus and an outermost sphere that was the domain of Aristotle divided the Universe into two realms -- the terrestrial and the what he called the Prime Mover (or Firmament in other references). The celestial -- with the moon's orbit as the boundary. Below the moon’s orbit Prime Mover rotated this outermost sphere with constant angular speed, was the terrestrial realm. This realm was composed of four primordial 06 Handout 1 *Property of STI  [email protected] Page 3 of 11  SH1690 elements in this sequence: Earth, water, air, and fire. Objects in the this small circle, in turn, moves around the Earth along a bigger circular terrestrial realm moved naturally according to their material composition. path called the deferent. To account for the variation in the sun's speed At or above the moon’s orbit was the celestial realm, which consists of during its annual motion, the Greek astronomer Hipparchus refined this the fifth element called aether (or ether). Aristotle considered the model by considering that Earth was off-center or eccentric in the terrestrial matter to be ephemeral and undergoing decay, while ether deferent where the sun moved. was unchanging and perpetual. According to Aristotle, Earth is a sphere. He based this proposition on several observations. First, it is only at the surface of a sphere that all objects fall straight down. Second, the view of the constellations changes as one travels from north to south. Finally, the shadow of the Earth on the moon during a lunar eclipse was round. Aristotle’s model was based on the three types of terrestrial motion: natural, violent, and alteration. Natural motion is related to the tendency of an object to seek its natural place in the Universe. Heavy elements move toward Earth, while lighter ones move up. However, terrestrial objects can be compelled to move in unnatural ways by the application of a force. This motion is considered a violent motion. Aristotle considered vertical motion as natural and horizontal motion as violent. He also considered motion as a type of change or alteration. Alteration is the ability of an object to change. This change can be generation, corruption, or alteration in quality. Generation is “coming to be”, while corruption is “passing away”. Figure 7. Ptolemy’s geocentric model contains a deferent and epicycles, with the sun as the deferent and the planets on epicycles Generation of one object results in the corruption of another and vice Source: https://astro.unl.edu/naap/ssm/modeling.html versa. Aristotle wrote in his book, On Corruption and Generation, that “coming to be and passing away take place when a thing changes, from Around 140 AD, Ptolemy devised a more complex epicyclic model. He this to that, as a whole. Passing away takes place when nothing defined a point on the other side of the deferent’s center and called it the perceptible persists the thing, and the thing changes as a whole.” equant. The equant and the center of the Earth are equidistant from the center of the deferent. When viewed at the equant, the epicycle orbited The end of motion is for the object to be at rest. On the other hand, an Earth at a constant rate. In Ptolemy’s model, each planet has its epicycle object changing its shape from rectangular to circular is considered an and deferent. His model of the Universe survived for more than 14 alteration in quality. centuries. Ptolemy’s Model The Greeks not only knew that the Earth was round. They also knew the The Greek mathematician Apollonius, known in his time as “The Great circumference of Earth to be 25,000 miles. Eratosthenes measured it in Geometer,” introduced the idea of an epicycle to explain planetary 235 BCE using trigonometry and the knowledge of the sun's angle of motion. An epicycle is a circle on which a planet moves. The center of elevation at noon in Alexandria and Syene (now Aswan, Egypt). 06 Handout 1 *Property of STI  [email protected] Page 4 of 11  SH1690 The Heliocentric Model Before the heliocentric model carne about, Greek astronomer Philolaus initially proposed a pyrocentric model of the Universe. According to him, neither Earth nor the sun was the center of the Universe. Planets and heavenly bodies were supposed to move around a ‘fire” located at the center of the Universe. In 300 BCE, another Greek astronomer Aristarchus proposed the first heliocentric model of the Universe by considering Philolaus’ “central fire” as the center of the cosmos. In this model, the sun and the other known planets revolve around this “central fire”. Aristarchus also placed the other known planets at that time based on their distances from the sun. However, his theory did not last because of the general acceptance of the Ptolemaic model. Copernicus’s Model Nicolaus Copernicus asserted that the Earth spins on its axis every day and revolves around the sun just like the other planets; only the moon orbits the Earth. He maintained the concept of uniform circular motion and Ptolemy’s epicycles. He gave reasonable explanations for the variation of brightness of planets and their retrograde motions. However, his model had two major scientific flaws: (1) the absence of stellar parallax and (2) the lack of perceived motion of Earth. Stellar parallax is the apparent displacement of a Star because of a change in the observer’s perspective. The Copernican model was not initially accepted because of its inconsistencies with Aristotelian mechanics and inability to explain stellar parallax. Copernicus’s book, Revolutionibus Orbium Coelestium (On the Revolution of Celestial Orbs), contained his heliocentric theory and was published in 1543. The Birth of Modern Astronomy Figure 8. The Copernican model of the Solar System, the second heliocentric model, attributed to Aristarchus’ own After Copernicus's death, three astronomers -- Tycho Brahe, Galileo model Source: http://farside.ph.utexas.edu/teaching/301/lectures/node151.html Galilei, and Johannes Kepler -- made significant contributions to modern astronomy. Each of them had a different approach. Tycho Brahe was a Tycho Brahe’s Universe good collector of astronomical data. Kepler was a mathematician and Tycho Brahe was considered the last and the greatest astronomer pure theorist, while Galileo was an experimentalist. Their contributions before the invention of the telescope. At the age of 30, he established helped prove that Earth is indeed not the center of the Universe. his own astronomical observatory in Hven, located between Denmark and Sweden, under the patronage of Danish King Frederick Il. In his observatory, he accurately measured and recorded the positions of the Sun, the Moon, and the planets for 20 years. Realizing that his data did not fit into Ptolemy and Copernicus's models, he proposed his own 06 Handout 1 *Property of STI  [email protected] Page 5 of 11  SH1690 model of the Universe. In his Universe, the sun orbited the Earth, while the sun. The Ptolemaic model can account only for the crescent the other planets orbited the sun. phase of Venus, not the full range of phases he observed. Many stars are too faint to see by the naked eye became visible with his telescope. He observed that the Milky Way was simply made of individual stars. Even when viewed through the telescope, the stars still appeared to be points of light. This provided evidence that the stars were extremely far away and that observing stellar parallax is extremely difficult. The Solar System Today The Solar System is now viewed as consisting of eight planets, with the sun as its center, and the planets revolve around the sun while spinning about their axes. Furthermore, the solar system is made up of zones. The innermost zones are occupied by the terrestrial planets Mercury, Venus, Earth, and Mars. These planets are rocky, metallic, and comparatively small. The next zone is the asteroid belt, where leftover Figure 9. Brahe’s model combined both geocentric and heliocentric tendencies in his model of the Solar System, but rocks from the Solar System formation can be found. Beyond the the sun is still the center Source: http://www.polaris.iastate.edu/EveningStar/Unit2/unit2_sub3.htm asteroid belt is the gas giants' realm -- Jupiter, Saturn, Uranus, and Neptune. Beyond the orbit of Neptune lies the Kuiper Belt, which Galileo’s Astronomical Observations consists of small celestial bodies. Pluto, which used to be a planet, is History had claimed that Dutch lens maker Hans Lippershey accidentally now classified as a “dwarf planet.” In 2006, Pluto lost its status as a invented a refracting in 1608. Upon hearing of this invention without planet because it is incapable of clearing debris off its orbital having seen it, Galileo made his own telescope and aimed it at the skies. neighborhood. Aside from Pluto, four other dwarf planets are known The following are some of the things he saw with his telescope, all of today -- Ceres, Haumea, Makemake, and Eris -- although more are which greatly contradicted Ptolemy and Aristotle's models and provided being discovered each day. new data that supported the Copernican model. These findings were published in 1610. INTRODUCTION TO CELESTIAL MECHANICS The moon has mountains, valleys, and craters. This suggested that the moon is not so different from Earth, implying that something in The Celestial Sphere the celestial realm is barely distinguishable from objects that belong The Ancient Greeks considered Earth to be enclosed in a hollow sphere to the terrestrial realm. called the celestial sphere, where the stars, the sun, and other heavenly The surface of the sun has some blemishes, which are now called bodies are embedded. They thought that the heavens' motion was sunspots. This observation contradicted the Greek concept of the caused by the rotation of the celestial sphere about a fixed Earth. The sun as being a perfect celestial body. points where the Earth’s rotational axis cuts this sphere are called the Jupiter has four moons revolving around it. This showed that not all north celestial pole (NCP) and the south celestial pole (SCP). The heavenly bodies revolve around the Earth. Other centers of a celestial equator is the projection of the Earth’s equator in the celestial revolution are themselves revolving. sphere. Venus has phases similar to those of the moon. This suggested that the sun's light merely illuminates Venus and that it revolves around 06 Handout 1 *Property of STI  [email protected] Page 6 of 11  SH1690 The path that the sun appears to take around the celestial sphere is called the ecliptic. It is inclined 23.5° concerning the celestial equator. The two points on the ecliptic with the celestial equator's greatest distance are referred to as solstices. The sun is at its northernmost position above the celestial equator or, at its highest in the sky, is called the summer solstice. It Is sometimes called the June solstice because it happens on or near June 21. The day is longest, and night is shortest during the summer solstice. The winter or December solstice occurs when the sun is at its southernmost position or its lowest in the sky. This normally happens on or near December 21. The day is shortest while the night is longest during the winter solstice. The two points where the ecliptic intersects the celestial equator are known as equinoxes. At the equinoxes, Earth’s rotational axis is perpendicular to the line joining Earth and the sun. On those days, day and night are of equal duration. The autumnal equinox happens on or near September 22. The vernal or spring equinox happens on or near March 21. The ecliptic traces through a series of star fields are called constellations, as defined by the International Astronomical Union. This sequence of constellations is called the zodiac. These constellations are located where the sun crosses in the sky. Different sets of constellations are visible in Earth’s night sky at different times of the year. Precession of the Equinoxes The Earth is not a perfect sphere. It bulges a bit in the equator because of the combined effects of the pull of the moon and the sun and the rapid rotation of the planet. As a result, Earth’s axis changes direction over Figure 10. The celestial sphere simplified some time. Such a change in the orientation of the rotational axis of any Source: http://www.sites.hps.cam.ac.uk/starry/armillmaths.html rotating body is termed precession. Earth requires 26 000 years to complete one cycle of precession. A complete cycle of precession traces stars in the sky, noting their exact positions in latitudes and longitudes. out a cone. He compared their positions with those measured by Timocharis about 150 years earlier. Hipparchus noted that there was a two-degree shift in The Earth’s precession was historically called the equinoxes' precession the positions of stars. Later on, the Earth’s precession due to the moon's because the equinoxes' position was slowly and gradually changing gravitational pull and the sun was called lunisolar precession. concerning some background stars. Hipparchus of Nicaea (known today as Turkey) was credited for discovering the precession of the equinoxes. He is said to have made a catalog of 06 Handout 1 *Property of STI  [email protected] Page 7 of 11  SH1690 Planetary Motion As theorized by Newton, planets and other celestial bodies have their paths shaped like conic sections. But, Newton was not the first to do so. Before he added mathematics into it, this theory was first studied and proven by Johannes Kepler, a mathematician who helped the astronomer Tycho Brahe in his astronomical research. Kepler’s calculations were based on precise data on apparent planetary motions compiled by Brahe. Through trial and error, Kepler eventually discovered three (3) empirical laws, which are now known as the Laws of Planetary Motion. The First Law The first law is called the Law of Elliptical Orbits, or Law of Ellipses. It states that all orbital paths of planets take on an elliptical shape, with the sun as one of its focus. If we are to recall our geometry, an ellipse is a conic section wherein it has foci located in its semi-major (or semi-major) axis. The sum of the distances from a specific focus (focus A) to a planet Figure 11. The Earth’s precession as compared to a top in orbit (P) and from P to another focus (focus B) is the same for all points Source: https://scienceatyourdoorstep.com/2014/09/08/what-is-precession/ along the curve. The sun sits on focus A, while focus B is just space. Diurnal Motion and Annual Motion It takes 24 hours for the Earth to rotate about its axis from west to east. Because of this, an observer on Earth views the objects in the sky as if they are the ones moving, but in the opposite direction -- from east to west. The apparent daily motion of stars and other celestial bodies across the sky caused by the Earth’s rotation about its axis is called diurnal motion. Diurnal motion is responsible for the daily rising and setting of the sun and the stars. The Earth also revolves around the sun once a year. As a result, the sun also apparently changes position in the celestial sphere, moving each day about one degree to the east relative to the stars. This apparent motion of the sun caused by the Earth’s revolution around it is called annual motion. Annual motion accounts for the visibility of a zodiac constellation at a specific time of the year. Along with the tilt of the Earth’s axis, it is also responsible for the seasons. Our changing Figure 12 The Law of Elliptical Orbits in the diagram perspective causes the diurnal and annual motions as the Earth rotates Source: http://hyperphysics.phy-astr.gsu.edu about its axis and revolves around the sun. 06 Handout 1 *Property of STI  [email protected] Page 8 of 11  SH1690 The distance of each focus from the center of the ellipse is 𝑒𝑎, where 𝑒 is the orbit's eccentricity. An eccentricity is a dimensionless number that describes how close a shape is from becoming a circle. As an example, if a shape has an eccentricity 𝑒 = 0, then that shape is a circle. ECCENTRICITY VALUES 𝑒=0 Circle 1>𝑒>0 Ellipse 𝑒=1 Parabola 𝑒>1 Hyperbola Figure 13. The Law of Equal Areas in the diagram Table 1. The eccentricities of each conic section Source: https://eloisechen.files.wordpress.com/ The actual orbits of planets are fairly circular, with Mercury being the To explain the analogy, let’s turn to mathematics. In a small time interval most eccentric (𝑒 = 0.206). Dwarf planets, such as Pluto, are highly 𝑑𝑡, the sun's line (focus A) to the planet (P) turns through an angle 𝑑𝜃. elliptical. So, calculating in between the foci and the planet’s The area created during that orbital movement has a height 𝑟 and a base displacement can determine whether a planet is at aphelion or length of 𝑟 𝑑𝜃, which gives us the area perihelion. 1 Aphelion is the farthest distance of a planet to the sun, while the 𝑑𝐴 = 𝑏ℎ 2 perihelion is the closest distance of a planet to the sun. 1 = (𝑟 𝑑𝜃)(𝑟) 2 Newton showed that the only possible closed orbits for planets are either 1 circles or ellipses because a body acted on by a pulling (i.e., attractive) = 𝑟 2 𝑑𝜃. 2 force is proportional to 1/𝑟 2. Any open orbits, such as the trajectories of hypervelocity objects in space, have parabolic and/or hyperbolic orbits. The rate at which the area was created (swept out in Physics), 𝑑𝐴/𝑑𝑡, is called the sector velocity and is represented as The Second Law The second law is called the Law of Equal Areas, which states that all 𝒅𝑨 𝟏 𝟐 𝒅𝜽 = 𝒓. calculated areas resulting from a planet’s movement around the sun are 𝒅𝒕 𝟐 𝒅𝒕 all equal in values. The “pizza slice” analogy best explains this. As the point of a planet sitting really close to the sun, the perihelion creates a The sector velocity, which is our “pizza slice,” has the same value at all very wide triangle or pizza slice. The aphelion, on the other hand, points in the orbit. For example, if a planet is in perihelion, the 𝑟 is small, creates a very narrow pizza slice. They may look different, but they have and 𝑑𝜃/𝑑𝑡 is large. Likewise, 𝑟 is large while 𝑑𝜃/𝑑𝑡 is small at aphelion. the same areas. To check that Kepler’s second law follows Newton’s laws, let us express 𝑑𝐴/𝑑𝑡 in terms of a velocity vector (𝑣) to a planet 𝑃. The component of 𝑣 is perpendicular to the radial line, where 𝑣⊥ = 𝑣 sin 𝜙. 06 Handout 1 *Property of STI  [email protected] Page 9 of 11  SH1690 The displacement along the direction of 𝑣⊥ during 𝑑𝑡 is 𝑟 𝑑𝜃, so we also The third law is called the Law of Harmonies, which states that by have observing the motions of two (2) planets moving around the sun, therein 𝑑𝜃 lies a calculated ratio where the ratio of the squares of the planets’ 𝑣⊥ = 𝑟. periods (𝑇) is equal to the ratio of the cubes of the planets’ average 𝑑𝑡 distances (𝑟) from the sun. The ratio is 3:2. Mathematically, Using this relationship with the sector velocity equation, we get (𝒓𝟏 )𝟑 (𝑻𝟏 )𝟐 𝒅𝑨 𝟏 =. = 𝒓𝒗 𝐬𝐢𝐧 𝝓. (𝒓𝟐 )𝟑 (𝑻𝟐 )𝟐 𝒅𝒕 𝟐 However, in isolated cases such as a planet orbiting along its orbit at a Since this is also part of the circular motion, 𝑟𝑣 sin 𝜙 is the magnitude of certain focus, a body's period in a circular motion is proportional to the the vector product 𝑟 × 𝑣, which in turn is 1/𝑚 times the angular 3/2 power of the orbit radius. Newton was able to show that this same momentum of a planet to the sun relationship also holds for an elliptical orbit, with the orbit radius 𝑟 replaced by the semi-major axis 𝐴, 𝐿 = 𝑟 × 𝑚𝑣. 𝟑 Thus, 𝟐𝝅𝑨𝟐 𝑻=. 𝑑𝐿 √𝑮𝒎𝑺 =𝜏 𝑑𝑡 = 𝑟 × 𝐹. Since the planets orbit around the sun, we replaced the Earth’s mass with the sun’s mass. Take note that the period does not depend on the In this case, 𝑟 is the vector from the sun to the planet, and the force 𝐹 is eccentricity. This makes that all objects will move within the same period directed from the planet to the sun. These vectors always lie along the -- it’s just that they move at various speeds at various points in their orbits same line, and the vector product is zero (0). Thus, to do so. 𝑑𝐿 The Analemma = 0. 𝑑𝑡 When viewed from a fixed position on Earth, the sun does not occupy the Same position in the same time every day in a year. This is due to This conclusion does not depend on the inverse relationship of the the following reasons: separation distance and force. Angular momentum is conserved for any The Earth’s axis is tilted 23.5° from the plane of its orbit around the force that always acts along the line joining the body to a fixed point. sun; This force is called the central force. The Earth rotates about its axis once a day as it revolves around the sun once every 365.26 days. Conservation of angular momentum also explains why orbits lie in a The Earth’s orbit is elliptical. It moves fastest at the perihelion plane. The vector 𝐿 is always perpendicular to the vectors 𝑟 and 𝑣; since (around December) and slowest at the aphelion (around June). 𝐿 is constant in both magnitude and direction, 𝑟 and 𝑣 always lie on the same plane (i.e., the orbital plane). The Third Law 06 Handout 1 *Property of STI  [email protected] Page 10 of 11  SH1690 REFERENCES Bauer, W. & Westfall, G. D. (2016). General physics 1 (2nd ed.). Columbus, OH: McGraw-Hill Education. Belleza, R.V., Gadong, E.S.A., …, Sharma, M. PhD. (2016). General physics 1. Quezon City: Vibal Publishing House, Inc Case Western Reserve University (n.d.). Kepler’s second law revisited. Lifted and modified from http://burro.case.edu/Academics/Astr221/Gravity/kep2rev.htm Engineering ToolBox. (2005). Torsion of shafts. Retrieved from https://www.engineeringtoolbox.com/torsion-shafts-d_947.html Freedman, R. A., Ford, A. L., & Young, H. D. (2016). Sears and zemansky’s university physics (with modern physics) (13th ed.). Boston, MA: Addison-Wesley Halliday, D., Resnick, R. & Walker, J. (2014). Fundamentals of physics (5th ed.). New York: John Wiley and Sons, Inc Santiago, K. S., & Silverio, A. A. (2016). Exploring life through science series: Senior high school physical science. Quezon City: Phoenix Publishing House Figure 14. The diagram of the sun’s analemma in the sky Source: https://en.wikipedia.org/wiki/Analemma The plot of the sun's position as viewed from a fixed position on Earth at the same time every day in a year is called an analemma. The analemma of the Sun on Earth looks like figure 8, with one broader than the other. The broadness of the loop depends on the location of the fixed position of the observer. Important Constants Earth's Mass = 5.972 × 1024 kg Earth's Radius = 6,371 km Sun's Mass = 1.989 × 1030 kg Sun’s Radius = 695,508 km 𝟏 𝐀𝐬𝐭𝐫𝐨𝐧𝐨𝐦𝐢𝐜𝐚𝐥 𝐔𝐧𝐢𝐭 (𝐀𝐔) = 149, 597, 871 km ≈ 1.5 × 108 km 06 Handout 1 *Property of STI  [email protected] Page 11 of 11  SH1690 Source: http://physicalsciencetext.weebly.com/ THE LAWS OF PHYSICS I. Basics of Motion Motion is defined as the change in position over a given time. By definition, it means that everything behaves linearly, so the objects and scenarios involved are only moving along one (1) of the three (3) known axes that we use (x, y, z). An example would be a car driving along the straight highway, without any other cars interfering with its movement. c. Acceleration is the change in velocity per unit of time. Meaning, it shows how fast the object moves to reach that There are several formulas we use to show motion. But before we certain amount of velocity within a certain amount of time. go into that, we need to define a few key terms used in this topic: Negative acceleration values indicate that the objects are a. Distance is the total measured length between two (2) given slowing down. points. Simply put, if we use a simple racetrack, it is measured from Start to the Finish line. II. Formulas and Representation b. Displacement is the measured mean distance covered Speed between two (2) specified points. Displacement is a certain Speed (represented as 𝑠) is the change in distance (𝑑) per unit distance at any given two (2) points. So, going back to the time (𝑡), as seen in the diagram. track, the displacement could be from Start to the middle, middle to the Finish line, or it could also be the average 𝑑 distance between the Start and the Finish line, used 𝑠 = | |. interchangeably with distance. 𝑡 c. Time is the progression of events occurring between This shows that 𝑣 has a direct instances. We need to prove that there is a progression relationship with 𝑑, but is inversely between events, such as from walking to running, jogging from point A to B, and so on. 𝒅 proportional to 𝑡. Statistically speaking, distance is the dependent Speed, Velocity, and Acceleration variable, while time is independent. a. Speed is the scalar quantity defined as the total distance The formula also represents the covered per unit time. Simply put, if there is motion, there is speed. Speed does not and cannot have a negative value. 𝒔 𝒕 formula for instantaneous speed, the speed at any given instant in time. Sometimes, it is interchanged with velocity. b. Velocity is the measured speed measured between two (2) Average speed (𝑣𝑎𝑣𝑒 ), defined as the specified points. It is the measured speed at a certain distance overall change in displacement and time, is written as at any given instance in time. However, unlike speed, velocity is a vector quantity and requires direction. Negative velocity ∆𝑑 values indicate that the objects are moving in the opposite 𝑠𝑎𝑣𝑒 =. ∆𝑡 direction. Expanding the equation above, 07 Handout 1 *Property of STI  [email protected] Page 1 of 9  SH1690 𝒅𝒇 − 𝒅𝟎 Acceleration 𝒔𝒂𝒗𝒆 = , Acceleration (𝑎) also has the same 𝒕𝒇 − 𝒕𝟎 diagram as both speed and velocity, but where, with some minor differences. Velocity (𝑣) 𝑑𝑓 = final distance reached becomes the dependent variable. Acceleration’s formula is seen in its 𝒗 𝑑0 = initial (or starting) distance given definition. 𝑡𝑓 = final time achieved 𝑡0 = initial (or starting) time The formula for acceleration, which is the change in velocity per set time, is, 𝒂 𝒕 Velocity Velocity (𝑣) also has the same diagram ∆𝑣 as speed, but with some minor 𝑎= differences. Displacement (𝑥) replaces the distance, and velocity replaces 𝒙 which is also equal to, ∆𝑡 speed, as seen in the diagram. (𝒗𝒇 − 𝒗𝟎 ) 𝒂=. The formula for velocity, defined as the change in displacement per change in 𝒗 𝒕 (𝒕𝒇 − 𝒕𝟎 ) unit time, is ∆𝑥 The above equation is also the equation used in determining the 𝑣= average acceleration of the object. If there are no given changes in ∆𝑡 both time and velocity, however (i.e., no given initial and final values), which is also equal to, acceleration can still be equated as an instantaneous acceleration, (𝒙𝒇 − 𝒙𝟎 ) where 𝒗=. 𝒗 (𝒕𝒇 − 𝒕𝟎 ) 𝒂=. 𝒕 As seen in the formulas, it also represents average velocity, the overall velocity of the changing displacements, and time. If the acceleration value is negative, it means that the object is slowing If we want to compute only for a specific velocity in a specified time, down or decelerating. we get the instantaneous velocity (𝑣𝑖𝑛𝑠 ), as seen in the formula, Derived Equations By combining and manipulating the given equations, we get the 𝒙 following derived equations, 𝒗𝒊𝒏𝒔 =. 𝒕 Displacement (𝑥) 𝑣̅0 + 𝑣̅𝑓 By simple analysis, we can see that speed and velocity are almost 𝑥=( )𝑡 2 alike in some ways. A negative velocity simply indicates that the object is moving in the opposite direction. 07 Handout 1 *Property of STI  [email protected] Page 2 of 9  SH1690 1 Take note of the changes in value. If we are to plot them, each graph = 𝑎𝑡 2 2 looks very different. First, let us plot distance vs. time. 1 2 = 𝑣̅0 𝑡 + 𝑎𝑡 Distance versus Time (Displacement vs. Time) 2 1 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 2 Time Vs Distance (A) Velocity (𝑣) 60 50 40 𝑣 = 𝑣̅0 + 𝑎𝑡 Distance 40 30 = 𝑎𝑡 20 𝑣 2 = 𝑣̅0 2 + 2𝑎∆𝑥 20 10 Graphical Representation 0 Graphs are one way of showing the relationship between the given 0 1 2 3 4 5 6 variables in Physics. Each of the three (3) rectilinear motion values Time has its graphical presentation. This is done to represent the changes in the system. Given a car moving at a constant, rightward velocity of 10 m/s, the Graphing helps us identify which part of the system is experiencing a chart shows the results of the car from zero (0) to five (5) seconds, uniform and/or uniformly accelerated motion. A uniform motion is a The resulting chart shows that the slope is a constant, rising, straight motion that experiences zero changes to its system. For example, a line. Since time is independent of distance, it is plotted as the x-axis, ball rolling at 0.3 meters every second will remain to roll at this rate whereas distance is plotted along the y-axis. This shows that the car because nothing makes it change its velocity (that is, acceleration is does not experience any change in its velocity. The height of the slope zero). also determines the rate of change the object experiences. This makes the graph a uniform motion. However, if an object moves with a set value of acceleration, then the object’s velocity will gradually change every second (provided that the Now, if we have the same car, this time having the following values, given acceleration is constant). This makes the motion a uniformly Time (s) Distance accelerated one. As an example, if a ball is accelerating at a rate of 0.3 meters every second per second after being pushed to move, then 1 2 the ball rolls as follows: 2 8 3 18 Time (s) Acceleration Velocity Displacement (m/s2) (m/s) (m) 4 32 1 0.3 0.15 5 50 2 0.9 1.35 0.3 Notice that each given value has exponentially risen than the previous. 3 1.8 5.40 This shows that the car is now experiencing a change in velocity, and 4 3.0 15.0 07 Handout 1 *Property of STI  [email protected] Page 3 of 9  SH1690 now acceleration is present. Plotting it in a graph, the results are as follows, Time Vs Distance (C) 60 50 48 Time Vs Distance (B) 50 40 Distance 40 60 50 30 22 50 20 32 Distance 40 10 0 30 18 0 20 8 0 1 2 3 4 5 6 10 2 Time 0 0 1 2 3 4 5 6 Time Now, you may notice that the distance value decreases. It does not imply that there is a negative distance because there is no such thing! Remember, distance is a scalar value. So, to solve this, use the As seen previously, the graph's slope has also changed, from a difference between the highest distance value and the lowest distance constant, straight line into a curve. This curve represents the sudden value. Now, looking at the previous given, the car’s initial displacement change in the car’s velocity, represented by a positive acceleration. is at 50 units. As it travels, it travels from 50 to zero (0) units. By This makes the graph a uniformly accelerated motion. definition, ∆𝑥 = (𝑥𝑓 − 𝑥0 ). But, if we are to follow it, we shall get -50. If the car travels as shown in the table below, assuming that the car To correct the total displacement, let us use ∆𝑥 = |𝑥𝑓 − 𝑥0 |. Replacing started at 50 units, the values, Time (s) Distance ∆𝑥 = |𝑥𝑓 − 𝑥0 | 1 50 = |0 − 50| 2 48 ∆𝒙 = 𝟓𝟎. 3 40 Now, adding it to the initial displacement value, we get a total distance 4 22 of 100 units. 5 0 Graphing values require analysis. If the results you got are different When plotted against the graph, it will show that the car shows from the others, you either have faulty data or interpreted the scenario negative acceleration, with the car moving slowly at first, then speeds wrong. Let us try an example. Suppose we have that same car up rapidly until it stops, as seen below. Take note that acceleration traveling a certain distance from its initial displacement of 80 units. values vary depending on use. This time, it has a random changing velocity due to it having acceleration changes, as seen on the next page. 07 Handout 1 *Property of STI  [email protected] Page 4 of 9  SH1690 Time (s) Distance Σ𝑥 = |𝑥0 + ∆𝑥2−1 + ∆𝑥3−2 + ∆𝑥4−3 + ∆𝑥5−4 + ∆𝑥6−5 | 1 80.00 = |80 + (92.5 − 80) + (110 − 92.5) + (65 − 110) + (35 − 65) + 0| = |80 + 12.5 + 17.5 + (−45) + (−30) + 0| 2 92.50 = |110| + |−75| 3 110.00 = 110 + 75 4 65.00 = 𝟏𝟖𝟓 𝐮𝐧𝐢𝐭𝐬. 5 10.00 Velocity versus Time 6 10.00 Velocity-time graphs are graphs meant to interpret changes for acceleration values. As discussed earlier, acceleration is the change As seen on the table, there are instances when the values remain of velocity per unit time. Therefore, if there is a substantial change in constant, such as the distance value of 10 units against the five (5) an object’s velocity within a certain amount of time, the object is said and six (6) second marks. It is interpreted as that the car is in a state to be accelerating. of rest or stationary for that time being. Presenting it in graph form, we get the values, Let us once again use the car as an example in this scenario. Provided below is the table for the car’s distance-time values, Time Vs Distance (D) Time (s) Distance 110.00 1 10 120.00 92.50 100.00 80.00 2 20 65.00 Distance 80.00 3 30 60.00 35.00 35.00 40.00 4 40 20.00 5 50 0.00 0 2 4 6 8 Time The graph of this table will be a steady, straight rise. Previously, we have learned that straight line rising steadily represents an object moving at a constant velocity. Presenting it in graph form, we get, As seen above, when the car revved up after one (1) second, the graph rises gradually at two (2) seconds, reaching its peak at three (3) seconds, which slopes down at four (4) to five (5) seconds. The sloping down of the graph means that the Grab car is experiencing negative, constant velocity. The car slows down and stops as it reached the five (5) second mark in the given scenario. Computing for the total distance, including the initial displacement of 80 units, we get 185 units. 07 Handout 1 *Property of STI  [email protected] Page 5 of 9  SH1690 Converting into a graph, we get, Time Vs Velocity (A) 12 10 10 10 10 10 Velocity 10 Time Vs Velocity (B) 8 15 6 10 Velocity 10 8 4 6 2 4 5 2 0 1 2 3 4 5 0 Time 1 2 3 4 5 Time In terms of acceleration, the car experiences zero (0) acceleration The graph shows a steady increase in velocity over a certain period. because the velocity values are equal on all levels, hinting that the car That is why acceleration is evident. Using example D, plotting the is moving steadily. But, if we are to make a velocity-time graph from velocity table, we get, the values as seen in the table below, Time (s) Velocity Time (s) Distance 1 0.00 1 2 2 10.00 2 8 3 15.00 3 18 4 -40.00 4 32 5 -30.00 5 50 6 0.00 Plotting it for a velocity-time table, we now get the following values, In graph form, we get, Time (s) Velocity 1 2 2 4 3 6 4 8 5 10 07 Handout 1 *Property of STI  [email protected] Page 6 of 9  SH1690 Time Vs Velocity 20.00 10.00 15.00 0.00 0.00 Velocity 0.00 1 2 3 4 -30.00 5 6 -20.00 -40.00 -40.00 -60.00 Time If we are now to combine the graphs, we get the graph seen at the right. As seen from the three graphs at the right, the time vs. acceleration graph shows no slope. That is because, in terms of calculus, adding a slope for an already constant value shall render the graph discontinuous or broken. Since acceleration is constant in value, if there is an evident change in velocity, there is no need to add a slope. Computing for the values using the graph, we use computation for the area. First, trace the acceleration graph. Then, match the values of the acceleration graph to get the velocity graph. Then, finally, match the velocity values to graph the distance graph. To compute for velocity using the time-acceleration graph, 𝐴 = 𝑙𝑤, where velocity is the Area (𝐴), the acceleration is the (𝑙), and time is the width (𝑤). Computing for the graph values is easy. If you are given a time- acceleration graph (i.e., the graph full of rectangular formations and no slant and curved lines), use the following equations: 1. To determine the velocity value of each point, use, 𝑣0 = the initial velocity (i.e., the velocity before the required 𝑣 = 𝑣0 + 𝑎𝑡, velocity) where, 𝑎 = the acceleration value on the graph 𝑣 = the required velocity 𝑡 = the time 07 Handout 1 *Property of STI  [email protected] Page 7 of 9  SH1690 2. To determine the displacement value of each point, use, Height (𝑦) 𝟏 1 𝒚𝒇 = 𝒚𝟎 + 𝒗𝟎 𝒕 − 𝒈𝒕𝟐 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 , 𝟐 2 Velocity (𝑣) where, ̅𝒇 = 𝒗 𝒗 ̅𝟎 + 𝒈𝒕 𝟐 𝑥 = the required displacement ̅𝒇 = 𝒗 𝒗 ̅𝟎 𝟐 + 𝟐𝒈∆𝒚 𝑥0 = the initial displacement (i.e., the displacement before the required displacement) References: Bauer, W., & Westfall, G. D. (2016). General physics 1 (2nd ed.). 𝑣0 = the initial velocity (i.e., the velocity before the required Columbus, OH: McGraw-Hill Education. velocity) Bauer, W., & Westfall, G. D. (2016). General physics 1 (2nd ed.). 𝑎 = the acceleration value on the graph Quezon City: Abiva Publishing House, Inc. Bautista, D.C. (2013). Science impact: Integrated science (3rd ed.). 𝑡 = the time Antipolo City: Academe Publishing House, Inc. III. Free-Fall Belleza, R.V., Gadong, E.S.A., …, Sharma, M. PhD. (2016). General Suppose a man went skydiving, jumping off his King Air 12,000 feet physics 1. Quezon City, Vibal Publishing House, Inc. above the ground, going from zero to 175 m/s within a short amount CHED. (2016). Displacement, time, average velocity, instantaneous of time. This is an example of free-fall, another case of rectilinear velocity. Retrieved 2017, February 13 from Teach Together: motion. CHED K-12 Curriculum Sharing Site: http://teachtogether.chedk12.com/teaching_guides/view/82 Free-fall is a motion wherein only gravity is the only acting influence Elert, G. (2016). Graphical representation of data. Retrieved October on the object. Since gravity pulls everything towards its center, it is 13, 2016, from The Physics Hypertextbook: safe to say that gravity is the only driving mechanism in an object’s http://physics.info/graphs/ vertical motion. The most common examples used in free-fall are the Freedman, R. A., Ford, A. L., & Young, H. D. (2011). Sears and balls being thrown upwards or dropped downwards in its classical zemansky’s university physics (with Modern Physics) (13th take. Acceleration here is nearly constant since gravity is the only ed.). Addison-Wesley. influence here. Known as lowercase 𝑔, the value of acceleration due Khan Academy. (n.d.). What is acceleration?. Retrieved 2017, to gravity is calculated to be 9.8 m/s 2, or 32.2 ft/s 2. Air resistance is February 7 from Khan Academy: “irrelevant” in classic free-fall, providing an “ideal” setup for its cases. https://www.khanacademy.org/science/physics/one- Formulas Used dimensional-motion/acceleration-tutorial/a/acceleration- The formulas used by free-fall are the same as those used by the article standard rectilinear motion, with slight differences. In this case, Santiago, K. S., & Silverio, A. A. (2016). Exploring life through science: acceleration 𝑔 is already given, whereas distance here is known as Senior High school physical science. Quezon City: Phoenix “height”, represented by 𝑦. Acceleration due to gravity, being a vector Publishing House, Inc. value, has a positive value when directed downwards. Also, graphing Somara, S. (2016). Motion in a straight line – Crash course physics free-fall values are the same as rectilinear motion. Maximum height is #1. Retrieved from YouTube: achieved if 𝑣̅𝑓 = 0. https://www.youtube.com/watch?v=ZM8ECpBuQYE 07 Handout 1 *Property of STI  [email protected] Page 8 of 9  SH1690 Texas Education Agency. (n.d.). Graphical representation of motion along a straight line. Retrieved 2017, February 13 from the Texas Gateway for Online Resources: https://www.texasgateway.org/node/4117 07 Handout 1 *Property of STI  [email protected] Page 9 of 9  SH1690 = √625 Kinematics: Two-Dimensional Motion 𝑽𝑹 = 𝟐𝟓 𝐦/𝐬. From 1D to 2D Solving for distance applies the same, As you can recall from the previous topic, we can present vectors within a Cartesian plane. As previously discussed as well, rectilinear 𝑅 = √𝑎 2 + 𝑏 2 motion is, simply put, motion along a straight line. So, projecting it = √402 + 302 within a Cartesian plane will only move either along the x-axis or the = √1600 + 900 y-axis. = √2500 In this lesson, we shall be dealing with motion when projected along 𝑹 = 𝟓𝟎 𝐦. two dimensions: x and y. As seen in the left image, there is a third Unfortunately, 2D motion requires an angle of displacement, arrow alongside the x and y axes. That arrow is called the resultant. It especially for resultant values. For this, we use 𝑆𝑂𝐻 − 𝐶𝐴𝐻 − 𝑇𝑂𝐴: represents the motion in both dimensions. Simply put, it 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑆𝑂𝐻, 𝑠𝑖𝑛𝑒 = simplifies the direction taken ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 by the object throughout its 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐶𝐴𝐻, 𝑐𝑜𝑠𝑖𝑛𝑒 = movement. ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 Suppose we have a car 𝑇𝑂𝐴, 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 = moving at 20 m/s, traveling at 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 40 m east, then travels at 30 However, take note that sine pertains to the angle moving from the y- m, moving at 15 m/s upon axis down to the x-axis. Cosine pertains to the angle moving from the turning north. How fast and x-axis up towards y, and tangent pertains to the incline’s slope (i.e., how far did the car travel? the resultant formed from the x and y values). So, solving for the angle, Solving this kind of problem 15 𝑐𝑜𝑠𝜃 = requires the following equation: 25 𝑐𝑜𝑠𝜃 = 0.6 𝜃 = cos −1 (0.6) 𝑉𝑅2 = 𝑉𝑥2 + 𝑉𝑦2 , 𝜽 = 𝟓𝟑. 𝟏𝟑° which is reminiscent of the Pythagorean Theorem, 𝑐 2 = 𝑎2 + 𝑏 2 , where, Projectile Motion Suppose we have a 65-kg man wearing an advanced prototype of a 𝑎 = 𝑉𝑥 3D Maneuver Gear (3DMG). He aimed his left and right reels at the 𝑏 = 𝑉𝑦 peaks of two abandoned stone towers 375 meters above the ground, 𝑐 = 𝑉𝑅. with the man standing at a building about 50 meters above the ground. Solving for the resultant velocity, The towers were 250 m away from the building. As the gear pulled him towards the two (2) towers at a rate of 56 m/s vertically, which changes 𝑉𝑅 = √𝑉𝑥2 + 𝑉𝑦2 after 4 seconds, as seen in the figure. What is the man’s vertical height as soon as he hit the 4-second mark? = √202 + 152 07 Handout 2 *Property of STI  [email protected] Page 1 of 3  SH1690 This setup is an example of Projectile Motion. Range The maximum horizontal displacement an (𝒙𝑴𝑨𝑿 ) object can attain Flight Time The total amount of time an object takes to (𝑻) travel from the initial point to its destination Always keep in mind these following notes: 1. 𝑥 gets maximum value if 𝜃 = 45° 2. When adding two elevation angles, keep in mind that if the sum is equal to 45°, then 𝑥 gets maximum distance. 3. 𝑥 is constant. To solve for it in projectile motion, use 𝑥 = 𝑣𝑥 𝑡, where 𝑣𝑥 = 𝑣 𝑐𝑜𝑠𝜃. The vertical component of projectile motion is a free-fall. Thus, the formulas for projectile motion are as follows, as seen below. Projectile motion is a free-falling motion where an object, usually First are the formulas for horizontal launch. Take note that 𝑔 (↓) = referred to as a projectile, flies over the air at a certain angle. The 9.8 m/s 2. Take note as well of the following values. object’s path is known as its trajectory. As it flies in the air, the velocity it gains has both horizontal and vertical components. This, along with the angle the ballistic is being thrown at, creates a parabolic trajectory. Horizontal Motion 𝑎𝑥 =0 There are six (6) important variables in projectile motion. Take note that the variable for velocity (𝑣̅ ) can be written without the bar (i.e., 𝑣) 𝑣̅𝑥 = 𝑣̅0𝑥 for convenience. 𝑥 = 𝑣̅0𝑥 𝑡 Launch The velocity of the object at the start of its ̅𝟎 ) Velocity (𝒗 trajectory. Vertical Motion 𝑎𝑦 =𝑔 Impact Velocity The velocity of the object as it comes closer to ̅𝒇 ) (𝒗 its destination. 𝑣̅𝑦 = 𝑣̅0𝑦 + 𝑔𝑡 1 Angle of The angle of an object’s initial velocity 𝑦 = 𝑣̅0𝑦 𝑡 + 𝑔𝑡 2 2 Elevation (𝜽) concerning (wrt) the range Maximum The highest altitude the object reaches along From these values, we get the following derived equations for a Height (𝒚𝑴𝑨𝑿 ) its trajectory horizontal launch, 2𝑦 𝑡=√ 𝑣𝑦 = √2𝑔𝑦 𝑔 07 Handout 2 *Property of STI  [email protected] Page 2 of 3  SH1690 𝑔 1 since such values of y are above the peak of the trajectory. Thus, if 𝑣0 = 𝑥 √ 𝑦 = 𝑔𝑡 2 𝑦 = 0, 2𝑦 2