Transcript lecture2_grav_rr

P113 Gravitation: Lecture 2
• Gravitation near the earth
• Principle of Equivalence
• Gravitational Potential Energy
2006: Assoc. Prof. R. J. Reeves
Gravitation 2.1
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
Altitude (km)
ag (m/s2)
0
9.83
8.8 km
9.80
400 km
8.70
35,700 km
0.225
• Question: If the object is let go is ag the acceleration towards
earth?
2006: Assoc. Prof. R. J. Reeves
Gravitation 2.2
Gravitation near the Earth - 2
• Consider an object sitting on scales on the earth’s surface.
FN = normal force
Fg = mag
• Along the r axis with positive outwards direction we have net
force
FN - mag
2006: Assoc. Prof. R. J. Reeves
Gravitation 2.3
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
Gravitation 2.4
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
Gravitation 2.5
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
Gravitation 2.6
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 2.7
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 2.8
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
Gravitation 2.9