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Ch. 6: Gravitation & Newton’s Synthesis

Ch. 6: Gravitation & Newton’s Synthesis. This cartoon mixes two legends: 1. The legend of Newton , the apple & gravity which led to the Universal Law of Gravitation . 2. The legend of William Tell & the apple.

bernard-ellison
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Ch. 6: Gravitation & Newton’s Synthesis

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  1. Ch. 6: Gravitation & Newton’s Synthesis This cartoon mixes two legends: 1. The legend of Newton, the apple & gravity which led to the Universal Law of Gravitation. 2. The legend of William Tell & the apple.

  2. It was verySIGNIFICANT & PROFOUNDin the 1600's whenSir Isaac Newtonfirst wrote Newton's Universal Law of Gravitation! This was done at the young age of about 30. It was this, more than any of his other achievements, which caused him to be well-known in the world scientific community of the late 1600's. • He used this law, along withNewton's 2nd Law(his 2nd Law!)plus Calculus, which he also (co-) invented, toPROVEthat the orbits of the planets around the sun must be ellipses. • For simplicity, we will assume in Ch. 6 that these orbits are circular. • Ch. 6 fitsTHE COURSE THEME OFNEWTON'S LAWS OF MOTIONbecause Newton used hisGravitation Law& his2ndLaw in his analysis of planetary motion. • His prediction that planetary orbits are elliptical is in excellent agreement with Kepler's analysis of observational data & with Kepler's empirical laws of planetary motion.

  3. When Newton first wrote the Universal Law of Gravitation, it was the first time, anyone had EVER written a theoretical expression (physics in math form) & used it toPREDICT something that is in agreement with observations! For this reason, Newton's Formulation of his Universal Gravitation Lawis considered THE BEGINNING OF THEORETICAL PHYSICS. • It also gave Newton his major “claim to fame”. After this, he was considered to be a “major leader” in science & math among his peers. • In modern times, this, plus the many other things he did, have led to the consensus that Sir Isaac Newton was the GREATEST SCIENTIST WHO EVER LIVED

  4. Newton’s Grave & a Monument to him are in Westminster Abbey in London, England.

  5. Inscription on Newton’s Gravestone: “Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of theplanets, the paths of comets, the tidesof the sea, the dissimilarities in rays of light,and, what no other scholar has previously imagined, the properties of thecolorsthus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25th December, 1642, and died on 20th March 1727. Newton’s Monument in Westminster Abbey.

  6. Sect. 6-1: Newton’s Universal Law of Gravitation • This is an EXPERIMENTAL LAW describing the gravitational force of attraction between 2 objects. • Newton’s reasoning: the Gravitational force of attraction between 2 large objects (Earth - Moon, etc.) is the SAMEforce as the attraction of objects to the Earth. • Apple story:This is likely not a true historical account, but the reasoning discussed there is correct. This story is probably legend rather than fact.

  7. If the force of gravity is being exerted on objects on Earth, What is the Origin of that Force? Newton’s realization was that the force must come from the Earth itself! He further realized that this same force must be what keeps the Moon in its orbit!

  8. The gravitational force on you is half of a Newton’s 3rd Law pair: Earth exerts a downward force on you, & you exert an upward force on Earth. When there is such a large difference in the 2 masses, the reaction force (the force you exert on the Earth) is undetectable, but for 2 objects with masses closer in size to each other, it can be significant.  This must be true from Newton’s 3rd Law! The Force of Attraction between 2 small masses is the same as the force between Earth & Moon, Earth & Sun, etc.

  9. By observing planetary orbits, Newton also concluded that the gravitational force decreases as the inverse of the square of the distance r between the masses. Newton’s Universal Law of Gravitation:“Every particle in the Universe attracts every other particle in the Universe with a force that is proportional to the product of their masses& inversely proportional to the square of the distance between them: F12 = -F21 [(m1m2)/r2] The direction of this force: Along the line joining the 2 masses  This must be true from Newton’s 3rd Law!

  10. Newton’s Universal Gravitation Law • This force is written as: G a constant, theUniversal Gravitational Constant Gis measured & is the same forALLobjects.Gmustbe small! • The measurement of G in the lab is tedious & sensitive because it is so small. • First done by Cavendish in 1789. • Modern version of Cavendish experiment: Two small masses are fixed at the ends of a light horizontal rod. Two larger masses are placed near the smaller ones. • The angle of rotation is measured. • Use Newton’s 2nd Law to get the vector force between the masses. Relate to angle of rotation & can extract G.  Cavendish Measurement Apparatus

  11. G =the Universal Gravitational Constant • Measurements Find, in SI Units: • The force given above is strictly valid only for: • Very small masses m1& m2 (point masses) • Uniform spheres • For other objects: We need integral calculus!

  12. TheUniversal Law of Gravitation is an example of an Inverse Square Law • The magnitude of the force varies as the inverse square of the separation of the particles • The law can also be expressed in vector form The negative sign means it’s an attractive force • Aren’t we glad it’s not repulsive?

  13. Comments  Force exerted by particle 1 on particle 2  Forceexerted by particle 2 on particle 1 21 This tells us that the forces form a Newton’s 3rd Law action-reaction pair, as expected. F21 = - F12 The negative sign in the above vector equation tells us that particle 2 is attracted toward particle 1

  14. Gravity is a “field force” that always exists between two masses, regardless of the medium between them. The gravitational force decreases rapidly as the distance between the two masses increases This is an obvious consequence of the inverse square law More Comments

  15. Example 6-1: Gravitational Force Between 2 People A 50-kg person & a 70-kg person are sitting on a bench close to each other. Estimate the magnitude of the gravitational force each exerts on the other.

  16. Example 6-2: Spacecraft at 2rE • Spacecraft at twice the Earth radius Earth Radius: rE = 6320 km Earth Mass: ME = 5.98  1024 kg m ME

  17. Example 6-2: Spacecraft at 2rE • Spacecraft at twice the Earth radius Earth Radius: rE = 6320 km Earth Mass: ME = 5.98  1024 kg FG = G(mME/r2) • At surface (r = rE) FG = weight = mg = G[mME/(rE)2] • At r = 2rE FG = G[mME/(2rE)2] = (¼)mg = 4900 N m ME

  18. Example 6-3: Force on the Moon Find the net force on the Moon due to the gravitational attraction of both the Earth & the Sun, assuming they are at right angles to each other. ME= 5.99  1024kg MM=7.35 1022kg MS = 1.99  1030 kg rME = 3.85  108 m rMS = 1.5  1011 m F = FME + FMS (vector sum!)

  19. F = FME + FMS (vector sum!) FME = G [(MMME)/(rME)2] = 1.99  1020 N FMS = G [(MMMS)/(rMS)2] = 4.34  1020 N F =[ (FME)2 + (FMS)2] = 4.77  1020 N tan(θ) = 1.99/4.34  θ = 24.6º

  20. Gravitational Force Due to a Mass Distribution • In can be shown, with integral calculus, that: The gravitational force exerted by a SPHERICALLY SYMMETRIC mass distribution of uniform density on a particle outside the distribution is the same as if the entire mass of the distribution were concentrated at the center. • So, assuming that the Earth is such a sphere, the gravitational force exerted by the Earth on a particle of mass m on or near the Earth’s surface is FG = G[(mME)/r2];MEEarth Mass, rEEarth Radius • Similarly, to treat the gravitational force due to large spherical shaped objects, it can be shown with calculus, that: 1) If a (point) particle is outside a thin spherical shell, the gravitational force on the particle is the same as if all the mass of the sphere were at center of shell. 2) If a (point) particle is inside a thin spherical shell, the gravitational force on the particle iszero. So, we can model a sphere as a series of thin shells. For a mass outside any large spherically symmetric mass, the gravitational force acts as though all the mass of the sphere is at the sphere’s center.

  21. Sect. 6-2: Vector Form of Universal Gravitation Law In vector form, The figure gives the directions of the displacement & force vectors. If there are many particles, the total force is the vector sum of the individual forces:

  22. Example: Billiards (Pool) • 3 billiard (pool) balls, massesm1 = m2 = m3 = 0.3 kg on a table as in the figure. Triangle sides: a = 0.4 m, b = 0.3 m, c = 0.5 m. Calculate the magnitude & direction of the totalgravitational force F on m1due to m2& m3. Note: Gravitational force is a vector, so we have to add the vectorsF21 & F31to get the vector F(using the vector addition methods of earlier). F = F21 + F31 Using components:Fx = F21x + F31x = 0 + 6.67  10-11 N Fy = F21y + F31y = 3.75  10-11 N + 0 So,F = [(Fx)2 + (Fy)2]½ = 3.75  10-11 N tanθ = 0.562, θ = 29.3º

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