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: Masses of the two objects 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 orbit? 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: 2pr = ???? m orbital period: T = 27.3 days = ???? sec orbital speed: v = (2pr)/T = ??? m/sec

The Moon’s Orbital Speed radius of orbit: r = 3.8 x 108 m Circumference: 2pr = 2.4 x 109 m orbital period: T = 27.3 days = 2.4 x 106 sec orbital speed: v = (2pr)/T = 103 m/sec = 1 km/s

The Moon’s Centripetal Acceleration The centripetal acceleration of the moon: orbital speed: v = 103 m/s orbital radius: r = 3.8 x 108 m centripetal acceleration: Ac = v2 / r = ???? m/s2

The Moon’s Centripetal Acceleration The centripetal acceleration of the moon: orbital speed: v = 103 m/s orbital radius: r = 3.8 x 108 m centripetal acceleration: Ac = v2 / r Ac = (103 m/s)2 / (3.8 x 108 m) = 0.00272 m/s2

The Moon’s Centripetal Acceleration At the surface of Earth (r = radius of Earth) a = 9.8 m/s2 At the orbit of the Moon (r = 60x radius of Earth) a =0.00272 m/s2 What’s relation between them?

The Moon’s Centripetal Acceleration At the surface of Earth (r = radius of Earth) a = 9.8 m/s2 At the orbit of the Moon (r = 60x radius of Earth) a =0.00272 m/s2 9.8 m/s2 / 0.00272 m/s2 = 3600 / 1 = 602 / 1

Bottom Line The Moon’s Centripetal Acceleration r 2r 3r 4r 5r 6r 60r g 1 4 9 16 25 36 3600 Figure 5-2 The force on a 0.1-kilogram mass at various distances from Earth. Notice that the force decreases as the square of the distance.

Bottom Line The Moon’s Centripetal Acceleration 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”)

Bottom Line The Moon’s Centripetal Acceleration 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 Ac ~ 1/r2 Gravity’s Inverse Square Law The acceleration due to gravity is inverse proportional to the square of the distance Ac ~ 1/r2 The gravity is inverse proportional to the square of the distance Fg = Fc = m Ac Fg ~Ac Fg ~ 1/r2

Bottom Line Fg ~ 1/r2 Gravity’s Inverse Square Law Gravity is reduced as the inverse square of its distance from its source increased Fg ~ 1/r2 r 2r 3r 4r 5r 6r 60r Fg 1 4 9 16 25 36 3600

Bottom Line Fg ~ 1/r2 Gravity’s Inverse Square Law Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

Gravity’s Inverse Square Law Bottom Line

Bottom Line Gravity’s Inverse Square Law 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 Gravity’s Inverse Square Law 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 Gravity’s Inverse Square Law Earth's surface 6.38 x 106 m Location Distance from Earth's center (m) Value of g (m/s2) Earth's surface 6.38 x 106 m 9.8 1000 km above 7.38 x 106 m 7.33 2000 km above 8.38 x 106 m 5.68 3000 km above 9.38 x 106 m 4.53 4000 km above 1.04 x 107 m 3.70 5000 km above 1.14 x 107 m 3.08 6000 km above 1.24 x 107 m 2.60 7000 km above 1.34 x 107 m 2.23 8000 km above 1.44 x 107 m 1.93 9000 km above 1.54 x 107 m 1.69 10000 km above 1.64 x 107 m 1.49 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 Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

Bottom Line Fg = G m1 m2 / d2 Law of Universal Gravitation Universal gravitation formula Fg = G m1 m2 / d2 Fg: gravitational force between objects G: universal gravitational constant m1: mass of one object m2: mass of the other object d: distance between their centers of mass Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

Bottom Line Law of Universal Gravitation d Fg m2 m1 p.83 p. 83: The attraction between these spheroidal friends depends on the distance between their centers.

Bottom Line Fg = G m1 m2 / d2 Law of Universal Gravitation 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 Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

Bottom Line Law of Universal Gravitation Example m1 m2 d F 2m1 3m2 2d Mass 1 Mass 2 Distance Relative Force m1 m2 d F 2m1 3m2 2d 3d 2m2 Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

Law of Universal Gravitation Example Mass 1 Mass 2 Distance Relative Force m1 m2 d F 2m1 2F 3m2 3F 6F 2d F/4 3d F/9 2m2 Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

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 Use Torsion balance (Metal thread, 6-foot wooden rod and 2” diameter lead sphere) Two 12”, 350 lb lead spheres The reason why Cavendish measuring the G is to “Weight the Earth” The measurement is accurate to 1% and his data was lasting for a century

Cavendish’s Experiment

Universal Gravitational Constant G = Fg d2 / m1 m2 = 6.67 x 10-11 N·m2/kg2 Fg = G m1 m2 / d2

Calculate the Mass of Earth G = 6.67 x 10-11 N·m2/kg2 Fg = G M m / r2 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·m2/kg2 Fg = G M m / r2 9.8 N = 6.67 x 10-11 N·m2/kg2 x 1 kg x M / (6.4 x 106 m)2 where M is the mass of Earth M = 6 x 1024 kg

Gravitational force is a Universal Gravitational Force G = 6.67 x 10-11 N·m2/kg2 Fg = G m1 m2 / d2 Gravitational force is a VERY WEAK FORCE

Universal Gravitational Force G = 6.67 x 10-11 N·m2/kg2 Gravity is is the weakest of the presently known four fundamental forces

∞ Universal Gravitational Force Force Strong Electro-magnetic Weak Gravity Strength 1 1/137 10-6 6x10-39 Range 10-15 m ∞ 10-18 m

Universal Gravitation Example Calculate the force of gravity between two students with mass 55 kg and 45kg, 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 45kg, and they are 1 meter away from each other Fg = G m1 m2 / d2 Fg = (6.67 x 10-11 N·m2/kg2)(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 m1 m2 / d2 Fg = (6.67 x 10-11 N·m2/kg2)(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 / r2 Weight Fg = m g Acceleration due to gravity g = G M / r2 Fg: gravitational force / weight G: univ. gravitational constant M: mass of Earth m: mass of the object r: radius of Earth g: 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 / r2 g = (6.67 x 10-11 N·m2/kg2)(5.98 x 1024 kg)/(6.37 x 106 m)2 = 9.83 m/s2

Universal Gravitation Example 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 104 kg, what is the acceleration due to gravity on asteroid B612?

Universal Gravitation Example 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 104 kg, what is the acceleration due to gravity on asteroid B612? g = G M / r2 g = (6.67 x 10-11 N·m2/kg2)(1.00 x 104 kg)/(20.0 m)2 = 1.67 x 10-9 m/s2

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 / r2 Fg ~ M / r2 Fg = 1000 N x 95 / (9.4)2 = 1075 N

Relative Weight on Each Planet

p. 85: The lunar rover would collapse under its own weight if used on Earth. (NASA)

Defined the World Isaac Newton’s Influence 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 Gravitational Field A gravitational field is a force field that surrounds massive objects

Earth’s Gravitational Field

Earth’s Gravitational Field 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

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 / r2) / m = G M / r2 F: weight of the object G: universal gravitational constant (6.67 x 10-11 N·m2/kg2) m: mass of the object M: mass of Earth (5.98 x 1024 kg) r: Earth’s radius (6.37 x 106 m)

Gravitational Field Inside a Planet

Gravitational Field Inside a Planet C r

Gravitational Field Inside a Planet Cancellation of gravitational force 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 P C r

Gravitational Field Inside a Planet B P Cancellation of gravitational force The gravity of area A and area B on P completely cancel out

Gravitational Field Inside a Planet C r Cancellation of gravitational force You are pulled only by the mass within this shell – below you 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 / r2 g = GM/r2 = GDV/r2 = GD(4/3)πr3/r2 = (4/3)GDπr g ~ r G: universal gravitational constant (6.67 x 10-11 N·m2/kg2) M: mass of Earth (5.98 x 1024 kg) D: density of Earth V: volume of Earth r: distance to the Earth’s center

Gravitational Field Inside a Planet a = g a = g/2 a = 0

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