Fields Tutorial 2 - Gravity Fields and Energy

The gravitational potential of a point is defined as:

 

The work done on a unit mass in moving it to that point from a point remote from all other masses.

 

In other words, this means the work done to move a unit mass from infinity to the point under consideration.

 

The zero point for gravitational potential is at infinity, so as we are moving towards the Earth, we are getting work out of the system.  Therefore gravitational potential is negative.  If we moved the object away to infinity, we would have to do a job of work on the object.  We work out gravitational potential with the formula:

 

 

 

[G = 6.67 10-11 Nm2 kg-2; M = mass of Earth or other planet; r = distance from centre.]

 

If we go back to our graph of gravitational field strength against distance we get:

 

 

We see that the gravitational potential is the area under the graph:

 

Question 1

The asteroid Ceres has a diameter of 785 km and a mass of 1.0 1020 kg.  Calculate:

(a)    the strength of the gravitational field;

(b)   the gravitational potential at the surface.

Answer

                  

 

If we plot V against r, we get the following graph:

 

 

From this graph we can see that the gradient gives us the gravitational field strength, g.  So we can say that:

 

 

 

We can go on to use the idea of gravitational potential to find an expression for the potential energy.  We have defined potential as energy per unit mass, so we can work out the total energy for any mass by multiplying the potential by the mass:

 

                                               

 

Another bear trap

 

Ep = mgDh is only true when we are very close to the Earths surface.  So do NOT use it for objects out in space.  Instead use

 

 

Worked Example

A satellite of mass 200 kg is to be moved from an orbit of 200 km above the Earth's surface to a an orbit of 400 km.  What work needs to be done?

(Radius of Earth = 6.37 106 m; mass of Earth = 6.0 1024 kg)

We need to calculate the gravitational potential energy at 200 km:

200 km = 0.2 106 m r = 6.57 106 m

Ep = - 6.67 10-11 Nm2 kg-2 6.0 1024 kg   200 kg = -1.22 1010 J

                                    6.57 106 m                                      

 

Now do the same for the 400 km:

 

400 km = 0.4 106 m r = 7.01 106 m

Ep = - 6.67 10-11 Nm2 kg-2 6.0 1024 kg   200 kg = -1.18 1010 J

                                            6.77 106 m        

 

Now we can work out the job of work we have to do to shift the satellite:

Work done = -1.18 1010 J - -1.22 1010 J = +0.04 1010 J =  +4.0 108 J

 

The plus sign tells us that a job of work has to be done.

 

 

Question 2

Some deep space cosmonauts land on a small planet.  They know that the mass of their craft is 500 103 kg and that to stop, they had to use 6.0 1011 J of energy.  They also worked out that the radius of the planet is 1500 km. 

(a)     Explain why the potential energy is - 6.0 1011 J.  Why is the sign negative?

(b)    What is the mass of the planet? 

(c)     What is its density? 

Answer

 

What a wretchedly bad piece of bad contrived science fiction!

 

 

Motion of Masses in Gravitational Fields

Newtons Laws of Gravitation can be used to explain the motion of planets and stars.  Much of modern space exploration uses the three hundred-year-old model.  Orbiting satellites are NOT doing gravity defiance acts; instead they are actually falling in a curved path towards the Earth all the time.  However they have a sufficient forwards velocity to miss the Earth all the time.  Since the gravity field is radial, the force acts at 90o to the direction of travel all the time.  Therefore the path is circular.  If we stopped gravity, the satellite would fly off tangentially into space in a straight line.  If we stopped the satellite, it would fall straight back to Earth.

 

 

For a satellite to be in a particular orbit, a particular velocity is required or a given distance.  Some satellites are placed so that they go in an easterly direction, completing one orbit each day.  They remain above one given point on the Earths surface, so are called geostationary.  This kind of satellite orbit is used in telecommunications.  Other satellites move in a polar orbit so that they can perform sweeps of the surface.  Spy satellites use a polar orbit.

 

When considering the motion of satellites in orbit, you have to know the rules of simple circular motion.  Click HERE to go back to Further Mechanics Tutorial 3 to revise these.

 

Useful formulae include:

 

Worked Example

A communications satellite is to be placed in a circular geostationary orbit.  What must its height and speed be?

This question seems to be remarkably lacking in information, but there is an answer.

 

Use Newtons Laws of Gravitation to solve this.  There is a single force acting on the satellite, gravitational attraction, so the satellite is acceleration all the time towards the centre of the Earth.

We need the satellite to be travelling at a sufficient forwards velocity so that it completes ONE orbit every 24 hours.  We need to work out the angular velocity before we can work out the linear speed.

w = 2pf.  We need to work out f.  f = ____1_____ = 1.16 10-5 Hz

                                                      24 60 60

 

w = 2 p 1.16 10-5 Hz = 7.27 10-5 rad/s.

Now we need to consider the centripetal acceleration:

 

a = w2r.  We also know that the acceleration is also given by g = -GM/r2

 

w2r = GM/r2 [We will ignore the minus sign.]

r3 = -GM = 6.67 10-11 N m2 kg-1 5.98 1024 kg = 7.55 1022 m3

                                    (7.27 10-5 Hz)2

 

r = 3 7.55 1022 m3 = 4.24 107 m

Now we can work out the forward velocity:

 

v = wr = 7.27 10-5 rad/s 4.24 107 m = 3100 m/s.

 

 

Question 3

The planet Mars has a diameter of 6800 km.  A satellite is in orbit 5000 km above the planet's surface travelling at a speed of 7100 m/s. 

(a)    How long does it take to orbit? 

(b)   What is the centripetal acceleration at this speed?

(c)    What is the acceleration due to gravity at this distance?

(d)   Will it remain in that orbit?  Mass of Mars = 6.42 1023 kg  

Answer

 

 

 

Equipotentials in Gravity Fields

In a uniform gravity field, we are familiar with the equation for potential energy:

 

 

We can rewrite the equation in terms of gravitational potential (energy per unit mass):

 

 

At a height of 10 m, the potential is 98 J kg-1; at 20 m, its 196 J kg-1, and so on.  We can show these potentials as lines.  Every point along these lines has the same potential, so they are called lines of equipotential.


 

 

If we move along a line, we do no work at all, because we have not moved any distance against the force of gravity.

 

Contours on a map are lines of equipotential.  They are marked in 10 metre intervals rather than 98 J kg-1 intervals, because the actual potentials would mean nothing to the map user.

 

In a radial field, the lines of equipotential are not equally spaced, as the energy per unit mass varies inversely.

 

 

A satellite orbiting the Earth along the -40 MJ kg-1 equipotential does zero work.  This is why it does not lose kinetic energy.  If it did, it would fall towards the Earth.

 

Question 4

What is the radius of the orbit of the satellite above?

How high is the orbit above the Earth's surface? (mass of the Earth = 5.98 1024 kg)

Answer