Earth Science PDF - Earth’s Measurements and Properties

Subject Code ES Earth Science Module Code 1.0 The Earth Lesson Code 1.1 Earth’s Measurements and Properties Time Frame 30 minutes Components Tasks Target 1. Compare the equatorial circumference and polar cir...

Subject Code ES Earth Science Module Code 1.0 The Earth Lesson Code 1.1 Earth’s Measurements and Properties Time Frame 30 minutes Components Tasks Target 1. Compare the equatorial circumference and polar circumference of the Earth 2. Account for the different Earth measurements based on the Earth's shape 3. Compute the quantities relevant to Earth, such as mass, diameter, circumference, volume, density, rotational speed and orbital speed Hook What is the Shape of the Earth? (from the words of Delfin C. Angeles) You might have encountered this question in your lower grade levels, “What is the shape of the Earth”? And most probably, most of you replied, “it is an oblate spheroid.”. What is meant by the term OBLATE SPHEROID? Spheroid The root word is sphere, which refers to a perfectly round object. Attached to the root word sphere is the suffix -oid. Putting together, the term spheroid simply means it is somewhat a sphere but not really a perfect sphere, since the Earth has a bulging equator and flattened poles. Oblate The term “oblate” needs more explanations. Reference lines are used for locating places on Earth. Reference lines parallel to the equator are called LATITUDES. The root word for latitude is the Latin word “latus/lateris”, which means “side”. Latitudes extend sideways. On the other hand, reference lines that extend from the North pole to the South Pole are called LONGITUDES. On a 3D globe, longitudes meet at the poles, but on a Mercator map, they appear as parallel lines extending from top to bottom. A perfectly round solid figure is a sphere. A deformed sphere is a spheroid. The deformation of a sphere may happen along either the longitude or the latitude, when the Earth deforms sideways (along the latitude) because of its rotational motion, we say that it is “off sideways”. “Ob-“ is a Latin prefix that means “on”. Since the deformity happens ON THE LATITUDE, the term is, thus, OBLATE. So, a sphere deformed on or along the latitude is an OBLATE SPHEROID. If in case that the deformity of a sphere happens along the LONGITUDE, then we have a spheroid whose shape is an OBLONG. Ignite ‘How big is the Earth?’ Modern technologies can help us answer this question with high accuracy, but do you know that you can try to measure the Earth’s circumference even without using sophisticated modern technologies? Earth Science Earth’s Measurements and Properties Page 1 of 9  The Greek Way Eratosthenes of Alexandria was the first person to devise a method of estimating the Earth’s circumference. He was captivated with geography, which brought him into his desire to map the world. But the challenge was, he first needed to measure the size of the Earth. Certainly, one cannot possibly walk all the way around the Earth to measure its actual size. As a mathematician, his simple method of Figure 1: Eratosthenes of Alexandria estimating the Earth’s circumference https://www.youtube.com/watch?v=EfZ2HZH5CkA consists of a combination of geometrical calculations and physical observations. Eratosthenes heard a fascinating observation from voyagers about a well in Syene (presently known as Aswan in Egypt). Accordingly, the whole bottom of this well is enlightened during noon time in summer solstice, which happens 21st of June annually. This observation indicates that the sun’s position was directly overhead in Syene during this period. Figure 2: Eratosthenes’ observation in summer solstice Figure 2 shows how Eratosthenes was https://www.youtube.com/watch?v=EfZ2HZH5CkA moved by his curiosity to investigate whether he would be observing the same phenomenon in Alexandria during summer solstice. At noon time, he measured the angle of the shadow of the stick (figure 3) that was formed, which is equal to 7.2 degrees or around 1/50 of a total circle. Figure 3: The angle of a shadow cast Eratosthenes could have easily calculated by the stick in Summer solstice the Earth’s circumference if he only knew the distance between Alexandria and https://www.youtube.com/watch?v=EfZ2HZH5CkA Syene, which is a challenging task to do. During the ancient time, distances from one city to another can be measured by a camel convoy. However, camels tend to roam and walk at different paces. To solve this, Eratosthenes hired land surveyors to measure the distance between Alexandria and Syene. It turned out that Alexandria and Syene are 5000 stadia apart. Earth Science Earth’s Measurements and Properties Page 2 of 9  Calculating the Earth’s Circumference 5000 stadia Figure 4: Eratosthenes’ Measurement of the Earth’s Circumference https://www.dummies.com/education/math/geometry/how-to-determine-the-earths-circumference/ 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 3600 = 5000 𝑠𝑡𝑎𝑑𝑖𝑎 7.20 3600 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = × 5000 𝑠𝑡𝑎𝑑𝑖𝑎 7.20 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 250,000 𝑠𝑡𝑎𝑑𝑖𝑎 During the time of Eratosthenes, the numerical equivalent of stadion (pl. stadia) as a unit of length varied. Here, we will use the itinerary stadion (used in measuring the distance of a journey) as our point of reference, which is equal to 157 meters. So, converting the value calculated by Eratosthenes: 157𝑚𝑒𝑡𝑒𝑟𝑠 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 250,000 𝑠𝑡𝑎𝑑𝑖𝑎 × 1 𝑠𝑡𝑎𝑑𝑖𝑜𝑛 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 39, 250, 000 𝑚𝑒𝑡𝑒𝑟𝑠 𝑜𝑟 39, 000 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = ~40, 000 𝑘𝑚 Eratosthenes was able to calculate the circumference of the Earth to be 40,000 km, which is almost accurate if we will base this numeric value to the known measurement of Earth’s equatorial circumference, which is 40,075 km. Equatorial vs Polar Circumference Based on the concept of oblate spheroid, the deformation causes a difference in the relative measurements of the Earth’s polar and equatorial circumference. The Earth’s rotation causes the bulging of the equator and flattening of the poles. The equatorial circumference is about 40,075 km, while the pole-to-pole (a.k.a. polar/meridional) circumference is less; only about 40,008 km round. Earth Science Earth’s Measurements and Properties Page 3 of 9  Self-Check #1 Based on the current value of the Earth’s circumference, calculate its radius and diameter. Show your complete solution. Navigate Calculating the Earth’s Mass For this part of our discussion, you will be introduced to one of the most famous equations in Physics, in able to calculate the Earth’s mass. The mass of the Earth may be calculated using Newton’s Law of Universal Gravitation, which states that: I. every object in the universe attracts every other object in the universe with a gravitational force; and II. the magnitude of the gravitational force between two objects is: 1. directly proportional to the product of their masses; and 2. inversely proportional to the square of the separation between their centers. Gravity/Gravitation: the property or tendency of a mass to attract another mass. Gravity/Gravitation: comes from the mass itself. By the fact it is a mass, it will attract another mass. Earth Science Earth’s Measurements and Properties Page 4 of 9  Newton’s Law of Universal Gravitation This expression combines Newton’s Law of Universal Gravitation and Newton’s Second Law of Motion: 𝐺 𝑚1 𝑚2 𝐹𝑔𝑟𝑎𝑣 = =𝑚𝑔 𝑑2 Where: Fgrav : gravitational force G : universal gravitational constant : (6.67 x 10-11 Nm2/kg2) “Newton-meter square per kilogram square” m1 : mass of the first object, which is more massive m2 / m : mass of the second object, which is less massive d : distance between the two masses g : acceleration due to gravity Where: Fgrav is directly proportional to the product of the two masses Fgrav is inversely proportional to the square of the distance between their centers By rearranging the equation to calculate the Earth’s mass, we will yield the equation: 𝐺 𝑚1 𝑚2 𝐹𝑔𝑟𝑎𝑣 = =𝑚𝑔 𝑑2 Earth Science Earth’s Measurements and Properties Page 5 of 9  Self-Check #2 Calculate the Earth’s mass based from the given equation. Note that the value of g on Earth is 9.8 m/s2. Show your complete solution. Calculating the Earth’s Volume We described the shape of the Earth to be an oblate spheroid, which is roughly spherical in shape. Since the Earth is almost spherical, you can calculate 4 its volume by using the formula for the volume of a sphere: 𝑉= 𝜋 𝑟3 , 3 where r is the Earth’s radius. Self-Check #3 Calculate the Earth’s volume based from the given equation. Show your complete solution. Calculating the Earth’s Density The Earth’s density, 𝜌 , can be calculated by dividing the value of the Earth’s 𝑚1 mass by its volume or simply, 𝜌=. 𝑉 Self-Check #4 Calculate the Earth’s density based from the given equation. Show your complete solution. Calculating the Earth’s Rotational and Orbital Speed Figure 5: Star trails at PSHS-Main Campus Grandstand Earth Science Earth’s Measurements and Properties Page 6 of 9  Figure 6: Star trails at Mt. Balagbag These photographs of star trails in PSHS-MC and at Mt. Balagbag show the apparent movement of stars in the night sky caused by the Earth’s rotation. Therefore, this is one of the evidences that will support the idea that the Earth is moving. Since the Earth is rotating, we can compute for the instantaneous velocity of a point on the surface of the Earth at a specific latitude by dividing the distance traveled in one day (in one rotation of the Earth) by the time it takes to complete a sidereal day (23.93 hours). Note that the sidereal day is the duration of time for the Earth to complete 360O rotation in space, with reference to ‘fixed’ stars. Self-Check #5 How fast is the Earth moving? Calculate the rotational speed at the specified latitudes. Show your solutions. Earth Science Earth’s Measurements and Properties Page 7 of 9  For the Earth’s orbital speed around the sun, this value can be calculated by dividing the circumference of the orbit by the Earth’s orbital period or: 2 𝜋 𝑟𝐸 𝑠= 𝑡 Where: s : orbital speed of the Earth rE : Earth’s orbital radius t : Earth’s orbital period (365.25 days) Self-Check #6 How fast is the Earth moving? Calculate the orbital speed of the Earth around the sun, with the known value of the average distance between the Sun and the Earth that is 149,597,870 kilometers (1 AU). Show your solution. Earth Science Earth’s Measurements and Properties Page 8 of 9  Problem Solving Activity. On a one whole sheet of paper, solve for the following problems given below. Show your complete solution and highlight your final answers. (graded, 15 points) 1. An astronaut namely Astroberns, who went to Jupiter has a mass of 189.60 lbs. If the radius of Jupiter is 6.99 x 107 m, and its mass is 1.90 x 1027 kg, what is: a. The acceleration due to gravity on Jupiter b. The weight of Astroberns on Jupiter c. The density of Jupiter 2. Calculate the circumference of Planet X from point A, where a shadow is present (θ = 10 degrees). Assuming that the sun is directly overhead at point B, its distance from point A is 900,410 meters. Express your final answer in km. 3. Calculate the percentage error of Eratosthenes’ measurement of Earth’s circumference. What are the possible sources of this error? Knot To summarize this lesson, the difference between the measurements of the Earth’s circumference and diameter is due to its shape: an oblate spheroid, which is characterized by having flattened poles and a bulging equator. Also, the centrifugal force due to Earth’s rotation causes outward bulge along the equator, causing the equatorial region to be larger in circumference and diameter. At present, geodesy helps us know more about the Earth’s measurement. It uses various land surveying techniques and mathematical calculations to accurately measure the Earth’s size and to map its shape. Modern geodetic methods such as global navigation satellite system and global positioning system (GPS) enable geodesists and other scientists to accurately study and measure the Earth’s surface. References: Tarbuck, E.J. & Lutgens, F.K. (2017). Eratosthenes and Earth’s Circumference. Earth Science (15th ed.,p639). Pearson Education Limited Sharp, Tim (2017, September 15). How Big is Earth? Retrieved from https://www.space.com/17638- how-big-is-earth.html Kuzoian, Alex (2017, December 27). How the Ancient Greeks Proved Earth was Round over 2,000 Years Ago. Retrieved from https://www.businessinsider.com/how-greek-eratosthenes- calculated-earth-circumference-2016-6 Prepared by: Carmina S. Dalida Reviewed by: Edman H. Gallamaso Position: Special Science Teacher III Position: Special Science Teacher-V Campus: Main Campus: SOCCSKSARGEN Region (SRC) Earth Science Earth’s Measurements and Properties Page 9 of 9