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P113 Gravitation: Lecture 2

P113 Gravitation: Lecture 2. Gravitation near the earth Principle of Equivalence Gravitational Potential Energy. Gravitation near the Earth - 1. The force on an object of mass m a distance r from the centre of the earth is

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P113 Gravitation: Lecture 2

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  1. P113 Gravitation: Lecture 2 • Gravitation near the earth • Principle of Equivalence • Gravitational Potential Energy 2006: Assoc. Prof. R. J. Reeves

  2. Gravitation near the Earth - 1 • The force on an object of mass m a distance r from the centre of the earth is • Newton’s second law tells us this force is related to a gravitational acceleration ag • Question: If the object is let go is ag the acceleration towards earth? 2006: Assoc. Prof. R. J. Reeves

  3. Gravitation near the Earth - 2 FN = normal force Fg = mag • Consider an object sitting on scales on the earth’s surface. • Along the r axis with positive outwards direction we have net force FN - mag 2006: Assoc. Prof. R. J. Reeves

  4. Gravitation near the Earth - 3 • The object is undergoing rotational motion about the centre of the earth with angular velocity  • Therefore the inward centripetal force of m2R is exactly given by • The normal force is what we would call the “weight” mg of the object • g is the “free-fall acceleration” which would be the measured acceleration of the object if it was let go. 2006: Assoc. Prof. R. J. Reeves

  5. Principle of Equivalence • Previously we had the equation and cancelled m from both sides. • In doing this division we have assumed that the gravitational mass in mag is the same as the inertial mass in the other two terms. • This assumption is the essence of Einstein’s Principle of Equivalence: gravity is equivalent to acceleration • Question: What would be some of the effects if this assumption was not valid? 2006: Assoc. Prof. R. J. Reeves

  6. Gravitational Potential Energy - 1 • Consider the gravitational force between two masses m1 and m2. • This force is zero only when r  • If r is not and the masses are free to move then they will approach each other. As they get nearer the force increases and correspondingly so does their kinetic energies. • Question: If we believe in conservation of energy, then how can we have an increasing kinetic energy from apparently nothing? • Answer: There must be another energy that is decreasing as the particle get closer - this is gravitational potential energy. 2006: Assoc. Prof. R. J. Reeves

  7. Gravitational Potential Energy - 2 • For r very large we expect small kinetic energy and thus also small and negative gravitational potential energy. • For r we have zero kinetic energy and thus zero gravitational potential energy. • We can derive an expression for the gravitational potential energy by considering the work needed to move a mass a small distance dr against the gravitational force. • If we start at separation R between the masses and move them until they are separated by then the total work is 2006: Assoc. Prof. R. J. Reeves

  8. Gravitational Potential Energy - 3 • Using the expression for F and noting that vectors F and dr are in opposite directions we get • Now doing the integration 2006: Assoc. Prof. R. J. Reeves

  9. Gravitational Potential Energy - 4 • The general rules of conservation of energy state that the change in potential energy between two positions is related to the work by Uf – Ui = – W • For our gravitational system Uf = U = 0. Therefore the gravitational potential energy of two masses separated by distance r is given by • If the object is a mass m, distance r from the centre of the earth, then its gravitational potential energy is 2006: Assoc. Prof. R. J. Reeves

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