2006: Assoc. Prof. R. J. Reeves Gravitation 2.1 P113 Gravitation: Lecture 2 Gravitation near the earth Principle of Equivalence Gravitational Potential Energy

2006: Assoc. Prof. R. J. Reeves Gravitation 2.2 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 a g Question: If the object is let go is a g the acceleration towards earth? Altitude (km)a g (m/s 2 ) km km ,700 km0.225

2006: Assoc. Prof. R. J. Reeves Gravitation 2.3 Gravitation near the Earth - 2 Consider an object sitting on scales on the earth’s surface. Along the r axis with positive outwards direction we have net force F N - ma g F N = normal force F g = ma g

2006: Assoc. Prof. R. J. Reeves Gravitation 2.4 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  2 R 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 Gravitation 2.5 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 ma g 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 Gravitation 2.6 Gravitational Potential Energy - 1 Consider the gravitational force between two masses m 1 and m 2. 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 Gravitation 2.7 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 Gravitation 2.8 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 Gravitation 2.9 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 U f – U i = – W For our gravitational system U f = 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