Universal Gravitation ChinSung Lin Isaac Newton 1643 1727
- Slides: 69
Universal Gravitation Chin-Sung Lin
Isaac Newton 1643 - 1727
Newton & Physics
Universal Gravitation Newton’s Law of Universal Gravitation states that gravity is an attractive force acting between all pairs of massive objects. Gravity depends on: q Masses of the two objects q Distance between the objects
Universal Gravitation - Apple
Universal Gravitation - Moon
Universal Gravitation - Moon
Universal Gravitation Newton’s question: Can gravity be the force keeping the Moon in its v orbit? v v v Newton’s approximation: Moon is on a circular orbit Even if its orbit were perfectly circular, the Moon would still be accelerated
The Moon’s Orbital Speed radius of orbit: r = 3. 8 x 108 m Circumference: 2 pr = ? ? m orbital period: T = 27. 3 days = ? ? sec orbital speed: v = (2 pr)/T = ? ? ? m/sec
The Moon’s Orbital Speed radius of orbit: r = 3. 8 x 108 m Circumference: 2 pr = 2. 4 x 109 m orbital period: T = 27. 3 days = 2. 4 x 106 sec orbital speed: v = (2 pr)/T = 103 m/sec = 1 km/s
The Moon’s Centripetal Acceleration The centripetal acceleration of the moon: orbital speed: orbital radius: v = 103 m/s r = 3. 8 x 108 m centripetal acceleration: Ac = v 2 / r = ? ? m/s 2
The Moon’s Centripetal Acceleration The centripetal acceleration of the moon: orbital speed: orbital radius: v = 103 m/s r = 3. 8 x 108 m centripetal acceleration: Ac = v 2 / r Ac = (103 m/s)2 / (3. 8 x 108 m) = 0. 00272 m/s 2
The Moon’s Centripetal Acceleration At the surface of Earth (r = radius of Earth) a = 9. 8 m/s 2 At the orbit of the Moon (r = 60 x radius of Earth) a =0. 00272 m/s 2 What’s relation between them?
The Moon’s Centripetal Acceleration At the surface of Earth (r = radius of Earth) a = 9. 8 m/s 2 At the orbit of the Moon (r = 60 x radius of Earth) a =0. 00272 m/s 2 9. 8 m/s 2 / 0. 00272 m/s 2 = 3600 / 1 = 602 / 1
Line. Acceleration The Moon’s. Bottom Centripetal r 2 r 3 r 4 r 5 r 6 r 60 r g 1 g 4 g 9 g 16 g 25 g 3600
Line. Acceleration The Moon’s. Bottom Centripetal If the acceleration due to gravity is inverse proportional to the square of the distance, then it provides the right acceleration to keep the Moon on its orbit (“to keep it falling”)
Line. Acceleration The Moon’s. Bottom Centripetal If the acceleration due to gravity is inverse proportional to the square of the distance, then it provides the right acceleration to keep the Moon on its orbit (“to keep it falling”) The moon is falling as the apple does !!! Triumph for Newton !!!
Bottom Line Law Gravity’s Inverse Square The acceleration due to gravity is inverse proportional to the square of the distance Ac ~ 2 1/r The gravity is inverse proportional to the square of the distance Fg = Fc = m Ac Fg ~ 1/r 2
Bottom Line Law Gravity’s Inverse Square Gravity is reduced as the inverse square of its distance from its source increased Fg ~ r Fg 1 2 1/r 2 r 3 r 4 r 5 r 6 r 60 r Fg 4 Fg 9 Fg 16 Fg 25 Fg 3600
Bottom Line Law Gravity’s Inverse Square Fg ~ 1/r 2
Bottom Line Law Gravity’s Inverse Square
Bottom Line Law Gravity’s Inverse Square Gravity decreases with altitude, since greater altitude means greater distance from the Earth's centre If all other things being equal, on the top of Mount Everest (8, 850 metres), weight decreases about 0. 28%
Bottom Line Law Gravity’s Inverse Square Astronauts in orbit are NOT weightless At an altitude of 400 km, a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface
Bottom Line Law Gravity’s Inverse Square Location Distance from Earth's center (m) Value of g (m/s 2) Earth's surface 6. 38 x 106 m 9. 8 6000 km above 1. 24 x 107 m 2. 60 1000 km above 7. 38 x 106 m 7. 33 7000 km above 1. 34 x 107 m 2. 23 2000 km above 8. 38 x 106 m 5. 68 8000 km above 1. 44 x 107 m 1. 93 3000 km above 9. 38 x 106 m 4. 53 9000 km above 1. 54 x 107 m 1. 69 4000 km above 1. 04 x 107 m 3. 70 10000 km above 1. 64 x 107 m 1. 49 5000 km above 1. 14 x 107 m 3. 08 50000 km above 5. 64 x 107 m 0. 13
Bottom Line Law of Universal Gravitation 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
Bottom Line Law of Universal Gravitation Universal gravitation formula Fg = G m 1 m 2 / d 2 F g: G: m 1: m 2: d: gravitational force between objects universal gravitational constant mass of one object mass of the other object distance between their centers of mass
Bottom Line Law of Universal Gravitation d m 1 Fg Fg m 2 p. 83
Bottom Line Law of Universal Gravitation Fg = G m 1 m 2 / d 2 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
Bottom Line Example Law of Universal Gravitation Mass 1 Mass 2 Distance Relative Force m 1 m 2 d F 2 m 1 m 2 d m 1 3 m 2 d 2 m 1 3 m 2 d m 1 m 2 2 d m 1 m 2 3 d 2 m 1 2 m 2 2 d
Law of Universal Gravitation Example Mass 1 Mass 2 Distance Relative Force m 1 m 2 d F 2 m 1 m 2 d 2 F m 1 3 m 2 d 3 F 2 m 1 3 m 2 d 6 F m 1 m 2 2 d F/4 m 1 m 2 3 d F/9 2 m 1 2 m 2 2 d F
Universal Gravitational Constant The Universal Gravitational Constant (G) was first measured by Henry Cavendish 150 years after Newton’s discovery of universal gravitation
Henry Cavendish 1731 - 1810
Universal Gravitational Constant Cavendish’s experiment q Use Torsion balance (Metal thread, 6 -foot wooden rod and 2” diameter lead sphere) q Two 12”, 350 lb lead spheres q The reason why Cavendish measuring the G is to “Weight the Earth” q The measurement is accurate to 1% and his data was lasting for a century
Cavendish’s Experiment
Universal Gravitational Constant G = Fg d 2 / m 1 m 2 = 6. 67 x 10 -11 N·m 2/kg 2 Fg = G m 1 m 2 / d 2
Calculate the Mass of Earth G = 6. 67 x 10 -11 N·m 2/kg 2 Fg = G M m / r 2 The force (Fg) that Earth exerts on a mass (m) of 1 kg at its surface is 9. 8 newtons The distance between the 1 -kg mass and the center of Earth is Earth’s radius (r), 6. 4 x 106 m
Calculate the Mass of Earth G = 6. 67 x 10 -11 N·m 2/kg 2 Fg = G M m / r 2 9. 8 N = 6. 67 x 10 -11 N·m 2/kg 2 x 1 kg x M / (6. 4 x 106 m)2 where M is the mass of Earth M = 6 x 1024 kg
Universal Gravitational Force G = 6. 67 x 10 -11 N·m 2/kg 2 Fg = G m 1 m 2 / d 2 Gravitational force is a VERY WEAK FORCE
Universal Gravitational Force G = 6. 67 x 10 -11 N·m 2/kg 2 Gravity is is the weakest of the presently known four fundamental forces
Universal Gravitational Force Strong Electromagnetic Strength 1 1/137 10 -6 6 x 10 -39 Range 10 -15 m ∞ 10 -18 m ∞ Weak Gravity
Universal Gravitation Example Calculate the force of gravity between two students with mass 55 kg and 45 kg, and they are 1 meter away from each other
Universal Gravitation Example Calculate the force of gravity between two students with mass 55 kg and 45 kg, and they are 1 meter away from each other Fg = G m 1 m 2 / d 2 Fg = (6. 67 x 10 -11 N·m 2/kg 2)(55 kg)(45 kg)/(1 m)2 = 1. 65 x 10 -7 N
Universal Gravitation Example Calculate the force of gravity between Earth (mass = 6. 0 x 1024 kg) and the moon (mass = 7. 4 x 1022 kg). The Earth-moon distance is 3. 8 x 108 m
Universal Gravitation Example Calculate the force of gravity between Earth (mass = 6. 0 x 1024 kg) and the moon (mass = 7. 4 x 1022 kg). The Earth-moon distance is 3. 8 x 108 m Fg = G m 1 m 2 / d 2 Fg = (6. 67 x 10 -11 N·m 2/kg 2)(6. 0 x 1024 kg) (7. 4 x 1022 kg)/(3. 8 x 108 m)2 = 2. 1 x 1020 N
Acceleration Due to Gravity Law of Universal Gravitation: Fg = G m M / r 2 Weight Fg = m g Acceleration due to gravity g = G M / r 2 Fg : G: M: m: r: g: gravitational force / weight univ. gravitational constant mass of Earth mass of the object radius of Earth acceleration due to gravity
Universal Gravitation Example Calculate the acceleration due to gravity of Earth (mass = 6. 0 x 1024 kg, radius = 6. 37 × 106 m )
Universal Gravitation Example Calculate the acceleration due to gravity of Earth (mass = 6. 0 x 1024 kg, radius = 6. 37 × 106 m ) g = G M / r 2 g = (6. 67 x 10 -11 N·m 2/kg 2)(5. 98 x 1024 kg)/(6. 37 x 106 m)2 = 9. 83 m/s 2
Universal Gravitation Example In The Little Prince, the Prince visits a small asteroid called B 612. If asteroid B 612 has a radius of only 20. 0 m and a mass of 1. 00 x 104 kg, what is the acceleration due to gravity on asteroid B 612?
Universal Gravitation Example In The Little Prince, the Prince visits a small asteroid called B 612. If asteroid B 612 has a radius of only 20. 0 m and a mass of 1. 00 x 104 kg, what is the acceleration due to gravity on asteroid B 612? g = G M / r 2 g = (6. 67 x 10 -11 N·m 2/kg 2)(1. 00 x 104 kg)/(20. 0 m)2 = 1. 67 x 10 -9 m/s 2
Universal Gravitation Example 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?
Universal Gravitation Example 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? Fg = G m M / r 2 Fg ~ M / r 2 Fg = 1000 N x 95 / (9. 4)2 = 1075 N
Relative Weight on Each Planet
Isaac Newton’s Influence Defined the World People could uncover the workings of the physical universe Moons, planets, stars, and galaxies have such a beautifully simple rule to govern them Phenomena of the world might also be described by equally simple and universal laws
Summary • • • Isaac Newton Universal gravitation – Apple and Moon? Moon’s centripetal acceleration Gravity’s inverse square law Law of universal gravitation Universal gravitational constant – Henry Cavendish Calculate the mass of Earth Weak gravitational force Acceleration due to gravity Newton’s influence
Gravitational Interaction Chin-Sung Lin
Gravitational Field Force Field A force field exerts a force on objects in its vicinity Magnetic Field A magnetic field is a force field that surrounds a magnet and exerts a magnetic force on magnetic substances Electric Field An electric field is a force field surrounding electric charges
Gravitational Field A gravitational field is a force field that surrounds massive objects
Earth’s Gravitational Field
Earth’s Gravitational Field q Earth’s gravitational field is represented by imaginary field lines q Where the field lines are closer together, the gravitational field is stronger q The direction of the field at any point is along the line the point lies on q Arrows show the field direction q Any mass in the vicinity of Earth will be accelerated in the direction of the field line at that location
Strength of Gravitational Field Strength of the gravitational field is the force per unit mass exerted by Earth on any object Gravitational Field g = Fg / m = (G m M / r 2) / m = G M / r 2 F: G: m: M: r: weight of the object universal gravitational constant (6. 67 x 10 -11 N·m 2/kg 2) mass of the object mass of Earth (5. 98 x 1024 kg) Earth’s radius (6. 37 x 106 m)
Gravitational Field Inside a Planet
Gravitational Field Inside a Planet P r C
Gravitational Field Inside a Planet Cancellation of gravitational force P If Earth were of uniform density, the gravity of the entire surrounding shell of inner radius equal to your radial distance from the center will completely cancel out r C
Gravitational Field Inside a Planet Cancellation of gravitational force The gravity of area A and area B on P completely cancel out A P B
Gravitational Field Inside a Planet Cancellation of gravitational force P You are pulled only by the mass within this shell – below you r C At Earth’s center, the whole Earth is the shell and complete cancellation occurs
Gravitational Field Inside a Planet Strength of the gravitational field is proportional to M / r 2 g = GM/r 2 = GDV/r 2 = GD(4/3)πr 3/r 2 = (4/3)GDπr g ~ r G: M: D: V: r: universal gravitational constant (6. 67 x 10 -11 N·m 2/kg 2) mass of Earth (5. 98 x 1024 kg) density of Earth volume of Earth distance to the Earth’s center
Gravitational Field Inside a Planet a=g a = g/2 a=0 a = g/2 a=g
Gravitational Field Inside a Planet 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 are pulled in every direction equally, so that the net force is zero and the gravitational field is zero
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