0 Universal Gravitation a. Universal gravitation: Since the gravitational force is experienced by all matter in the universe, from the largest galaxies down to the smallest particles, it is often called universal gravitation. b. Gravity is the way in which masses communicate with each other. Every mass in the universe reaches out to attract every other one, and every mass feels an attraction from every other one. c. General theory of relativity: Gravity rises from the warping of space and time. General theory of relativity proposed in 1916 by Albert Einstein explained these differences and provided a geometric explanation for gravitational phenomena, holding that matter causes a curvature of the space-time framework in its immediate neighborhood. d. Examples: projectiles, satellite, planets, galaxies, and clusters of galaxies are all influenced by gravity. 1 The Falling Apple a. Galileo s concept of inertia without an outside force, moving objects continue to move at constant speed in a straight line. b. Newton s first law of motion An object continues in its state of rest, or of motion in a straight line at constant speed, unless it is compelled to change that state by an outside unbalanced force. c. Isaac Newton s concept of gravity If an object undergoes a change in speed or direction, then a force is responsible. The falling apple changes its speed and the revolving moon changes its direction. Both changes require a force. Newton has the insight to see that the moon is falling toward Earth, just as the apple is. Both of them are pulled by Earth s gravity. Newton didn t discover the gravity but expanded Galileo s concept of inertia and discovered that gravity is universal. 2 The Falling Moon a. The falling moon: The moon falls around Earth because it falls beneath the straight line it would follow if no force acted on it. The moon is simply a projectile circling Earth under the attraction of gravity. b. Newton s cannon-ball illustration: The mountain top cannon is above Earth s atmosphere, so that air resistance would not impede the motion of the cannonball. If it were fire faster, its path would be less curved and it would hit Earth farther away. If the cannonball were fired fast enough, its path would become a circle indefinitely. It would be in orbit. c. Tangential velocity: Both the orbiting cannonball and the moon have a component of velocity parallel to Earth s surface. This tangential velocity is sufficient to ensure nearly circular motion around Earth rather than into it. d. The proof of Newton s hypothesis: Newton s test was to see if the moon s fall beneath its otherwise straight-line path was in correct proportion to the fall of an apple or any object at Earth s surface. i) The mass of the moon should not affect how it falls, just as mass has no effect on the acceleration of freely falling objects on Earth s surface. ii) How far the moon falls, and how far an apple at Earth s surface falls, should relate only to their respective distances from Earth s center. iii) If the distance of fall for the moon and the apple are in correct proportion, then the hypothesis that Earth s gravity reaches to the moon must be taken seriously. iv) The moon was already known to be 60 times farther from the center of Earth than an apple at Earth s surface. Mr. Lin 1
v) The apple will fall 4.9 m in its first second of fall. s = ½ gt 2 = ½ x 9.8 m/s 2 x (1 s) 2 = 4.9 m vi) The influence of the gravity should be diluted 1/(60) 2. 4.9 m x 1/(60) 2 = 0.0014 m = 1.4 mm vii) Using geometry, Newton calculate the falling distance between the circle of the moon s orbit and the straight line the moon otherwise would travel in one second. His value turned out to be about 1.4 mm. viii) But he was unsure of the distance between Earth and the moon, and whether or not the correct distance to use was the distance between their centers. At this time he hadn t prove mathematically that the gravity of the spherical Earth is the same as if all its mass were concentrated at its center. For this and other reasons, the discovery of universal gravitation was delayed for near 20 years. ix) The law of universal gravitation: All objects in the universe attract each other. 3 The Falling Earth a. Copernican theory of the solar system: The sun is the center of the solar system. Earth and the planets orbit the sun in the same way that the moon orbits Earth. The planets continually fall around the sun in close paths because they have tangential velocities. 4 Newton s Law of Universal Gravitation a. Newton s discovery: Newton didn t discover gravity. In stead, he discovered that the gravity is universal. Everything pulls everything in a beautifully simple way that involves only mass and distance. b. Universal gravitation formula: F = G m 1 m 2 / d 2 F: gravitational force between objects G: universal gravitational constant m 1 : m 2 : mass of one object mass of the other object d: distance between their centers of mass c. Universal gravitational constant: G was first measured by Henry Cavendish 150 years after Newton s discovery of universal gravitation. Cavendish s experiment: i) Use Torsion balance (Metal thread, 6-foot wooden rod and 2 diameter lead sphere) ii) Two 12, 350 lb lead spheres. iii) The reason why Cavendish measuring the G is to Weight the Earth. iv) The measurement is accurate to 1% and his data was lasting for a century. A simpler version was later developed by Philipp von Jolly. G = F d 2 / m 1 m 2 = 6.67 x 10-11 N m 2 /kg 2 d m sensitive balance flask of mercury M 6-ton lead sphere F Mr. Lin 2
d. Gravity is weak: The value of G tells us that the force of gravity is a very weak force. It is the weakest of the presently known four fundamental forces. (The other three are electromagnetic force and two kinds of nuclear forces.) Force Strong Electromagnetic Weak Gravity Strength 1 1/137 10-6 6x10-39 Range 10-15 m 10-18 m e. Weight: The force between you and Earth is your weight. Your weight depends on your mass and your distance from the center of mass of Earth. At the top of a mountain your weight is slightly less than at ground level because your distance from the center of Earth is greater and also the mountain top is farther away from higher density part of Earth. (The continental crust is less dense than the material it floats upon.) f. Mass of Earth: The mass of Earth can be calculated once the G is measured. The force that Earth exerts on a mass of 1 kg at its surface is 9.8 newtons. The distance between the 1-kg mass and the center of mass of Earth is Earth s radius, 6.4 x 10 6 m. Thus, 9.8 N = 6.67 x 10-11 N m 2 /kg 2 x 1 kg x m e / (6.4 x 10 6 m) 2, where m e is the mass of Earth and m e = 6 x 10 24 kg 5 Gravity and Distance: The Inverse-Square Law a. Inverse-square law: Just like electrical force between charges, spreading of light from a candle, and the weakening of radioactivity as distance from the source increases, gravity follows an inverse-square law. Gravity is reduced as the inverse square of its distance from its source increased. b. Gravity is always there: Though the gravity decreases rapidly with the distance, it never drop to zero. The gravitational influence of every object, however small or far, is exerted through all space. c. Calculate the gravity: Relative Mass 1 Relative Mass 2 Relative Distance Relative Force m 1 m 2 d F 2m 1 m 2 d 2F m 1 3m 2 d 3F 2m 1 3m 2 d 6F m 1 m 2 2d F/4 m 1 m 2 3d F/9 2m 1 2m 2 2d F 6 Universal Gravitation a. Why is Earth round? It is round because of gravitation. Earth has had attracted itself together before it became solid. Any corners of Earth have been pulled in so that Earth is a giant sphere. b. Perturbation: Planets deviation from their normal orbits is called perturbation. For example, the perturbation of Uranus leads to the discovery of Neptune. c. Newton s influence: Newton had demonstrated that by observation and reason, people could uncover the workings of the physical universe. How profound it is that all the moons and planets and stars and galaxies have such a beautifully simple rule to govern them. It provided hope that other phenomena of the world might also be described by equally simple and universal laws. Mr. Lin 3
7 Gravitational Field a. Force Field A force field exerts a force on objects in its vicinity. A magnetic field is a force field that surrounds a magnet and exerts a magnetic force on magnetic substances. The pattern of the iron filings shows the strength and direction of the magnetic field at different points in the space around the magnet. Where the filings are closest together, the field is strongest. An electric field is a force field surrounding electric charges. A gravitational field is a force field that surrounds massive objects. b. Earth s gravitational field is represented by imaginary field lines. Where the field lines are closer together, the gravitational field is stronger. The direction of the field at any point is along the line the point lies on. Arrows show the field direction. Any mass in the vicinity of Earth will be accelerated in the direction of the field line at that location. c. Strength of the gravitational field is the force per unit mass exerted by Earth on any object. Gravitational Field: g = F m = G M R 2 F: weight of the object G: universal gravitational constant (6.67 x 10-11 N m 2 /kg 2 ) m: mass of the object M: mass of Earth (5.98 x 10 24 kg) R: Earth s radius (6.37 x 10 6 m) Near the surface of Earth, g = 9.8 m/s 2. The g is a vector quantity, for it has both magnitude (strength) and direction. Gravitational field vector g and acceleration due to gravity g have the same magnitude and their units are equivalent: m/s 2. The numerical value of g at Earth s surface depends only on the mass of Earth and its radius. So, all free falling objects have the same acceleration. If Earth had a different mass or radius, g at its surface would have a different value. As R increases, g decreases. By knowing the mass and radius of any planet, we can calculate the acceleration due to gravity at the surface of that planet. d. Calculate the gravitational field strength: Relative Earth s Mass Relative Earth s Radius Gravitational Field Strength M R g 2M R 2g M R/2 4g 2M R/2 8g M 2R g/4 2M 2R g/2 8 Gravitational Field Inside a Planet a. Gravitational Field Inside Earth: gravitational field exists both inside and outside Earth. b. Imaginary tunnel from the North Pole to the South Pole: If you started falling from the North Pole end, you d fall and gain speed all the way down to the center, and then overshoot and lose speed all the way to the South Pole. You d gain speed moving toward the center, and lose speed moving away from the center. Without air drag, the trip would take nearly 45 minutes. The gravitational field strength is steadily decreasing as you continue toward the center. At the center of Earth, you a = g a = g/2 a = 0 a = g/2 a = g Mr. Lin 4
are pulled in every direction equally, so that the net force is zero and the gravitational field is zero. There is no acceleration as you whiz with maximum speed past the center of Earth. If you did not grab the edges at either end, you would oscillate back and force, approximating simple harmonic motion. Each round trip would take about 90 minutes, similar to an Earth satellite in close orbit about Earth. c. Cancellation of gravitational force: If Earth were of uniform density, half way to the center gravitational force would be exactly half that at the surface. Cancellation is of the entire surrounding shell of inner radius equal to your radial distance from the center. Half way down, for example, all mass in the shell that surrounds you contributes zero gravitational force on you. You are pulled only by the mass within this shell below you. It is this Earth mass you use in Newton s equation for gravitation. At Earth s center, the whole Earth is the shell and complete cancellation occurs. 9 Weight and Weightlessness a. Weight: The force of gravity, like any force, cause acceleration. We are almost always in contact with Earth. For this reason, we think of gravity primarily as something that presses us against Earth rather than as something that accelerates us. The pressing against Earth is so called weight. b. Weight and motion: Steady Motion Accelerate Upward Accelerate Downward Broken Rope Normal Weight Greater Weight Less Weight No Weight c. Weightless: Weightless is not because of the missing of gravity. Rather than define your weight as the force of gravity that acts on you, it is more practical to define weight as the force you exert against a supporting floor. The condition of weightless is not the absence of gravity; rather, it is the absence of a support force. Astronauts in orbit are without a support force and are in sustained state of weightlessness. 10 Ocean Tides a. Ocean tides: Newton showed that the ocean tides are caused by the differences in the gravitational pull of the moon on opposite sides of Earth. It is because the gravitational force is weaker with increased distance. b. Two high tide per day: i) Both Earth and the moon are circling about their combined center of mass center of mass of the Earth-moon system. A point about 3/4 of the way from Earth s center to its surface. ii) The ocean nearest the moon is pulled upward toward to moon., while the main body of Earth is pulled toward the moon also away from the ocean on the far side. It is because Earth is closer to the moon than the far-side ocean is. iii) Earth s water gets slightly elongated at both ends. c. Moon contributes more than sun: i) Not because the pull from the moon is stronger, but because the difference in pulls on either sides of Earth is greater for the closer moon. Mr. Lin 5
ii) The sun s contribution: The sun s pull on Earth is about 180 times stronger than the moon s pull on Earth. Because the sun s great distance from Earth, there is not much difference. That is, The small difference in the solar pull on opposite sides of Earth only slightly elongates Earth s shape and produce tidal bulges less than half those produced by the moon. iii) The sphere of Earth: is torn into an elliptical shape by difference in gravitational forces exerted by moon. iv) Spring Tides and neap tides: The tides are extra high when the moon and sun are lined up (called spring tides) because the pulls add and an the two tides due to the moon and sun overlap. When the moon and sun are at right angles to each other, the high tides of the sun overlaps the low tides of the moon (called neap tides). d. Earth caused tides on the moon: i) The solid moon is slightly football shaped. The center of gravity is slightly displaced from its center of mass. Both lie along the moon s long axis. ii) Whenever the long axis of the moon is not lined up toward Earth, Earth exerted a small torque on the moon. iii) This will twist the moon toward aligning with Earth s gravitational field. iv) This is why the moon always shows us its same face. v) This tidal lock is also working on Earth. e. High and low tides: Average high tide is about 1 meter above the average surface level of the ocean. Average low tide is about 1 meter below the average surface level of the ocean. They are six hours apart. f. Tides do not occur at the same time: While Earth spinning, the moon revolves about Earth and appear at the same position in our sky every 24 hours and 50 minutes. That s the period of two high tides. That s why tides do not occur at the same time everyday. g. Inverse cube relation: Newton deduced that the difference in pulls decreases as cube of the distance between the centers of Earth and the moon. h. Size of the body: The size of the tides also depends on the size of the body. So, there is no significant difference in the moon s pull on lake, human fluid, etc. i. Tilt of Earth s axis: Thought the opposite tidal bulges are equal, because of the tilt of Earth s axis, the daily high tides are never the same. j. Ignored factors: interfering land masses, tidal inertia, and friction with the ocean bottom. k. Comet Shoemaker-Levy 9(SL9): SL9 was in pieces ranging in size up to 2 kilometers in diameter, and is believed to have been pulled apart by Jupiter's tidal forces during a close encounter in July 1992. These fragments collided with Jupiter's southern hemisphere over a period of time between July 16 and July 22, 1994, at a speed of approximately 60 kilometers per second (37 miles per second). 11 Tides in Earth and the Atmosphere a. Earth tides: Earth is not rigid but molten liquid covered by a thin solid and pliable crust. The moon-sun tidal forces produce Earth tides as well as ocean tides. Twice each day the solid surface of Earth rises and falls by as much as 25 cm. b. Atmospheric tides: Tidal effects in the ionosphere produce electric currents that alter the magnetic field that surrounds Earth. These are magnetic tides which regulate the degree to which cosmic ray penetrate into the lower atmosphere. The penetration affects the composition of our atmosphere, which in turn changes the behavior of living things. Mr. Lin 6
12 Black Holes a. Two processes in the sun: i) Gravitation: will crunch all solar materials. ii) Thermonuclear fusion: Hydrogen bomb-like reaction, will blow solar materials outwards. b. Generation of black hole: i) A star which is 2-3 times more massive than sun. ii) Run out of fusion fuel (hydrogen). iii) Gravitational force dominates and starts to collapse. iv) The collapse ignites fusion of the nuclear ashes (helium) and fuses them into carbon. v) The star (such as the sun) will expand and extend beyond Earth s orbit and swallows Earth. (Red Giant) vi) All the helium are burned out and start to collapse and die out. No longer give out heat and light. vii) Gravitational collapse starts, caves on itself and this process never stops. No empty space and infinite density. Nothing can get back out, and even light cannot get out, and can not be observed. (Black hole) viii) Thought near the black hole the gravity is huge, gravity has no change beyond its original radius. 13 Problems about Universal Gravitation a. Calculate the force of gravity between Earth (mass = 6.0 x 10 24 kg) and the moon (mass = 7.4 x 10 22 kg). The Earth-moon distance is 3.8 x 10 8 m. b. The moon falls beneath it s tangential straight path 1.4 mm each second. If the moon orbited three times as far from Earth, how far will it fall each second? c. The planet Saturn has a mass that is 95 times as massive as Earth and a radius that is 9.4 times Earth s radius. If an object is 1000 N on the surface of Earth, what is the weight of the same object on the surface of Saturn? Mr. Lin 7
d. In The Little Prince, the Prince visits a small asteroid called B612. If asteroid B612 has a radius of only 20.0 m and a mass of 1.00 x 10 4 kg, what is the acceleration due to gravity on asteroid B612? e. At what distance from Earth s center must a spacecraft be in order to experience the same gravitational attraction from both Earth and the moon when directly between the two? (M E = 5.98 x 10 24 kg, M M = 7.35 x 10 22 kg, and d E-M = 3.84 x 10 8 m). f. An object is falling into a tunnel which connects the North Pole and the South Pole. a) Describe the gravity change of the object along the path from A to D. b) Identify where can we find the maximum speed? c) What will happen if nobody will catch the object on the other side of Earth. Why? (Ignore the friction, air drag and technical feasibility.) g. There are gravitational attraction among the sun, the moon and Earth. Calculate the net force pulling on Earth during a) a new moon, b) a full moon and c) a first quarter moon. (M Moon = 7.35 x 10 22 kg, M Earth = 5.98 x 10 24 kg, M Sun = 1.99 x 10 30 kg, d E-M = 3.84 x 10 8 m, d E-S = 1.5 x 10 11 m) 1 st Quarter Moon New Moon EARTH Full Moon Mr. Lin 8