Sect. 13.4: The Gravitational Field

Gravitational Field A Gravitational Field , exists at all points in space. If a particle of mass m is placed at a point where the gravitational field is, it experiences a force: The field exerts a force on the particle. The gravitational field is defined as Gravitational field = Gravitational Force experienced by a “test” particle placed at that point divided by the mass of the test particle. –The presence of the test particle is not necessary for the field to exist The source particle creates the field

The gravitational field vectors point in the direction of the acceleration a particle would experience if it were placed in that field. Figure The magnitude is that of the freefall acceleration, g, at that location. The gravitational field describes the “effect” that any object has on the empty space around itself in terms of the force that would be present if a second object were somewhere in that space

Sect. 13.5: Gravitational Potential Energy

The gravitational force is conservative Recall Ch. 8 discussion of conservative forces: Only for conservative forces can a potential energy U be defined & Total Mechanical Energy is Conserved for Conservative Forces ONLY.  Just as in Ch. 8, define the change in Gravitational Potential Energy associated with a displacement of a mass m is as the negative of the work done by the gravitational force on m during the displacement. That is: F(r) is the Gravitational Force. For a mass m in the Earth’s gravitational field,

Doing this integral: This gives As always, the reference point where the potential energy is zero is arbitrary. Usually choose it at r i  , so (1/r i )  0 and U i  0. This gives For example, as a particle moves from point A to B, as in the figure  its gravitational potential energy changes by  U =

Notes on Gravitational Potential Energy in the Earth’s Gravity We’ve chosen the zero for the gravitational potential energy at r i   where the gravitational force is also zero. This means that U i = 0 when r i   or that This is valid only for r ≥ R E & NOT for r < R E –That is, it is valid outside the Earth’s surface but NOT inside it! –U is negative because of the choice of U i

Gravitational Potential Energy in Earth’s Gravity The figure is a graph of the gravitational potential energy U versus r for an object above the Earth’s surface. Note that the potential energy goes to zero as r approaches infinity

For any two particles, masses m 1 & m 2, the gravitational potential energy function is The gravitational potential energy between any two particles varies as 1/r. (Recall that the force varies as 1/r 2 ) The potential energy is negative because the force is attractive & we’ve chosen the potential energy to be zero at infinite separation. Some external energy must be supplied to do the positive work needed to increase the separation between 2 objects –The external work done produces an increase in gravitational potential energy as the particles are separated & U becomes less negative Gravitational Potential Energy: General Discussion

The absolute value of the potential energy for mass m can be thought of as the binding energy of m. (The energy of binding of m to the object which is attracting it gravitationally). Consider two masses m 1 & m 2 attracting each other gravitationally. If an external force is applied to m 1 & m 2 giving them an energy larger than the binding energy, the excess energy will be in the form of kinetic energy of m 1 & m 2 when they are at infinite separation. Binding Energy

Systems with Three or More Particles For systems with more than two masses, the total gravitational potential energy of the system is the sum of the gravitational potential energy over all pairs of particles. Because of this, gravitational potential energy is said to obey the superposition principle. Each pair of particles in the system contributes a term to U total. Example; assume 3 particles as in the figure. The result is shown in the equation The absolute value of U total represents the work needed to separate the particles by an infinite distance

Sect. 13.6: Energy Considerations in Satellite Motion

Energy and Satellite Motion Consider an object of mass m moving with a speed v in the vicinity of a large mass M –Assume that M >> m as is the case for a small object orbiting a large one. The total mechanical energy is the sum of the system’s kinetic and potential energies. Total mechanical energy: E = K +U In a system in which m is bound in an orbit around M, can show that E must be less than 0

Consider an object of mass m moving in a circular orbit about a large mass M, as in the figure. The gravitational force supplies the centripetal force: F g = G(Mm/r 2 ) = ma = m(v 2 /r) Multiply both sides by r & divide by 2: G[(Mm)/(2r)] = (½)mv 2 Putting this into the total mechanical energy & doing some algebra gives: = Energy in a Circular Orbit

The total mechanical energy is negative in for a circular orbit. The kinetic energy is positive and is equal to half the absolute value of the potential energy The absolute value of E is equal to the binding energy of the system

It can be shown that, for an elliptical orbit, the radius of the circular orbit is replaced by the semimajor axis, a, of the ellipse. This gives: The total mechanical energy is negative The total energy is conserved if the system is isolated Energy in an Elliptical Orbit

Consider an object of mass m projected upward from the Earth’s surface with an initial speed, v i as in the figure. Use energy to find the minimum value of the initial speed v i needed to allow the object to move infinitely far away from Earth. E is conserved (E i = E f ), so, to get to a maximum distance away (r max ) & then stop (v = 0): (½)mv i 2 - G[(M E m)/(R E )] = -G[(M E m)/(r max )] We want v i for r max    the right side is 0 so (½)mv i 2 = G[(M E m)/(R E )]. Solve for v i = v escape This is independent of the direction of v i & of the object mass m! Escape Speed from Earth

The result for Earth can be extended to any planet The table gives escape speeds from various objects. Note: Complete escape from an object is not really possible –Gravitational force extends to infinity so some force will always be felt no matter how far away you get This explains why some planets have atmospheres and others do not –Lighter molecules have higher average speeds and are more likely to reach escape speeds Escape Speed, General