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Fields & Effects

 

Gravitational Fields 4

 

gravitational potential U

field strength g

relation between g and U

energy in orbits

 

 

 

Gravitational Potential - U

 

 

definition of gravitational potential

 

 

The potential U at a point in a gravitational field is defined as being numerically equal to the work done by the field in bringing a unit mass from infinity to the point.

 

By definition, the potential at infinity is zero.

 

gravitational potential - equation #1

 

where,

 

U is the gravitational potential at a point

W is the work done in bringing a mass m from infinity to that point.

 

 

The units for gravitational potential (work/mass) are Jkg-1 .

 

In the treatment of escape velocity(gravitational fields 2) the work done in moving a mass m from the surface of the Earth to infinity was given by:

 

escape velocity #6

 

Now if we make the mass unity(m = 1 kg), the energy difference between the 1 kg mass on the surface of the Earth and at infinity (zero potential) is W .

 

However, since the highest potential is zero at infinity, all potential energies relative to this level are less than zero (ie negative).

 

Our 1 kg mass on the Earth therefore has a potential of -W .

 

If UE is the potential on the surface of the earth, then:

 

gravitational potential equation #3

 

substituting for W from above(remembering that m = 1kg),

 

gravitational potential - equation #4

 

Therefore in the general case, the potential Ur at a point a distance r away from a large mass M is given by:

 

gravitational potent U at a distance r from a mass M

 

 

graph of gravitational potential U against radial displacement r

 

 

The shape of the curve of Ur against r is of the type y = - x-1 , which is a reflection of the curve y = x-1 about the x-axis.

 

 

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Gravitational Field Strength - g

 

The field strength g at a point in a gravitational field is defined as the force on unit mass at the point.

 

gravitational field strength

 

So the unit of gravitational field strength is Nkg-1 .

 

From Newton's Law of gravitation, the force on a mass m as a result of a mass M at a distance r is given by:

 

making the mass m unity (1 kg),

 

field strength equation #2

 

 

variation of gravitational field strength with radial distance

 

 

The shape of the curve of g against r is of the type y = x-2 .

 

Near the surface of the Earth, the value of g is considered to be constant at 9.8 Nkg-1.


The number should look familiar. This is also g , the acceleration due to gravity 9.8 ms-2 .

 

The two g's are exactly the same, with the same dimensions.

 

g - gravitational field strength equation #1

 

 

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Derivation of the relation between g and U

 

Consider a particle of mass m in a gravitational field.


In the absence of any applied force, the mass would be attracted to the major body producing the field.

 

Let the mass be held in position by a force F , acting in the opposite direction to the field direction.

 

Now if the force F moves the mass a very small distance δx against the field, the work done is given by:

 

(work = force x distance force moves)

 

g - U relation equation #1

 

assuming that the force F is constant.

 

When the mass is static, the net force is zero. Forces are balanced. Since forces are vector quantities, the minus sign signifies opposite direction.

 

g - U relation equation #2

 

Substituting for F into our original equation,

 

g - U relation equation #3               (i               

 

By definition, gravitational potential U is given by:

 

gravitational potential - equation #1

 

So δU , the increase in U, is given by:

 

g - U relation equation #4

 

Substituting for δW from equation (i ,

 

g - U relation equation #5

 

cancelling the mass m gives,

 

g - U relation equation #6

 

rearranging, to make -g the subject,

 

g - U relation equation #7

 

In the limit, g - U relation equation #7b , therefore

 

g - U relation equation #8

g - U relation equation #9

 

 

U - g graph

 

 

From the graph it can be seen that the gravitational field strength g at a radius r is equal to minus the value of the gradient of the gravitational potential U .

 

Note that g is positive because the value of dU/dr itself is negative. Multiplying a negative by a negative gives a positive.

 

 

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Energy in Orbits

 

The energy Er of a satellite of mass m in orbit, of radius r around a large body of mass M, is the sum of the satellite's PE and KE respectively,

 

kinetic and potential energy for a circular orbit of radius r

 

This equation can be simplified by elimenating v2 .

 

Recalling the equation describing the circular motion of the satellite,

 

orbit energy equation #2

 

cancelling r

orbit energy equation #3

 

and making v2 the subject.

 

orbit energy equation #4

 

We can now substitute for v2 in the initial energy equation:

 

orbit energy equation #5

 

So the total energy Er of the satellite in its orbit is given by:

 

orbit energy equation #6

 

 

graph of orbital PE and KE against radius

 

 

NB

 

1. The total energy of the satellite is always negative.

 

2. The PE component of the energy is twice as large in absolute terms as the KE component.

 

Also, for any particular circular orbit with radius r , the individual values of kinetic and potential energies are constant.

 

By contrast, with elliptical orbits the values of potential and kinetic energies are not constant. They vary such that when one is large the other is small and vice versa.

 

It must be remembered that the sum of potential and kinetic energies is always constant for a particular orbit.

 

 

 

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