9.2.A - Gravity


Gravity

Gravity is a natural phenomenon by which physical bodies attract each other with a force proportional to their mass. In everyday life, gravity is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped. Gravity causes matter to coalesce and remain intact, thus accounting for the existence of the Earth, the Sun, and most of the large objects in the universe. Gravity is responsible for keeping the Earth and the other planets in their orbits around the Sun; for keeping the Moon in its orbit around the Earth; for the formation of tides; for natural convection, by which fluid flow occurs under the influence of a density gradient and gravity; for heating the interiors of forming stars and planets to very high temperatures; and for various other phenomena observed on Earth.

Gravity is one of the four fundamental forces in physics, along with electromagnetism, and the nuclear strong force and weak force. Modern physics describes gravitation using the general theory of relativity by Einstein, in which it is a consequence of the curvature of spacetime governing the motion of inertial objects. The simpler Newton's law of universal gravitation provides an accurate approximation for most physical situations and is the focus of study in this section of work.

In this unit you will learn about:

  • Newton's law of universal gravitation and those factors that affect the strength of gravitational force
  • The gravitational field, how it is defined and measured
  • Weight as another way of describing graviational force
  • Gravitational potential energy and those factors that affect it


Gravitational Force

The ideas specified by Newton in what has now become the Law of Universal Gravitation were first put forward in his monumental work on classical mechanics, Philosophiae Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687.

Any two objects with mass and separated by a
distance will experience a force between them.

This law states that any two objects with mass separated by a certain distance exert a gravitational force of attraction on each other. The direction of the force is along the line joining the objects. The magnitude of the force is directly proportional to the product of the masses of the objects and inversely proportional to the square of the distance between them (inverse square law).

F1 and F2 are the same as they represent an action-reaction pair from Newton's Third Law. F1 is the force acting on F2 and F2 is the force acting on F1. Both F1 and F2 have the same magnitude but act in opposite directions.

Mathematically these relationships can be expressed as follows:

Equation - Calculating the Force of Gravity

F

G

m1

m2

r

force between the masses

universal gravitation constant (6.673 x 10-11)

mass of heavier object

mass of lighter object

distance between the centres of each mass

newtons (N)

N m2 kg-2

kilograms (kg)

kilograms (kg)

metres (m)

You should use this equation when you are trying to calculate the force of gravity between any two objects that have mass with their centres separated by some distance. Examples include calculating the force of gravity between you and the Earth or the force of gravity between the Sun and the Earth.

 

The relationships between the force and the masses and distance can be shown graphically as follows:

This graph shows how the force varies between a 1 kg mass and another increasing mass when they are separated by a distance of 1 m. The force is directly proportional to the mass, meaning that they have a linear relationship that passes through the origin. This graph shows how the force varies between two 1 kg masses when the distance between them increases. The force is inversely proportional to the distance squared. The force drops away at a much higher rate than the distance is increased due to the square in the denominator.


It should be apparent that the size of the force between 1 kg masses is extremely small. This is why you don't feel a force of attraction to the person next to you. It is much too small to notice!

You should also note that the force of gravity between objects can never be zero. The graph of y=1/x2 is asymptotic so mathematically the force can only be zero at infinitiy. This means that a spacecraft can never escape the force of the Earth's gravity. In reality though, because of the shape of the graph, the force drops very quickly to negligible values so the force of attraction at large distances may as well be zero.



Science in a Minute Video: Newton's Law of Universal Gravitation. Note that this video treats the relationship
between variables qualitatively rather than quantitatively but it's a good introduction.

 

Gravitational Fields

The gravitational field surrounding a large
mass such as the Earth. The red
concentric circles represent lines of
equipotential where the magnitude of
g is constant.

In physics, fields are described as being a region of influence around whatever it is producing the field.

Gravitational fields are no exception. Every mass produces a gravitational field that infuences other masses in the field by way of an attractive force. Let's consider the Earth's gravitational field and a deep space probe moving in the field away from Earth. The force on the space probe is decreasing as it moves away from the Earth so we could say that the Earth's gravitational field strength is also decreasing.

Qualitative descriptions of phenomena such as fields are not adequate in physics. Physicists seek to describe the world mathematically, so a description of exactly how much force is experienced by a precisely known mass is needed to more effectively describe the gravitational field strength at any point in the field.

Gravitational field strength is defined as the force per unit mass acting on a test mass in the gravitational field of another mass.

Said another way, the gravitational field strength is defined by the force and direction that a 1 kg mass would experience

M1 produces a gravitational field, the strength of which can be
determined by the force on a 1 kg test mass m2.

in that gravitational field. Note that the force of gravity is always attractive so the direction of the gravitational field is always towards the object producing the field.

Field lines show the direction of the force experienced by a mass in the field. Since gravity is always attractive, the field lines always point towards the object producing the gravitational field. This is the unusual thing about the gravitational field. Other fields such as electric and magnetic fields can be attractive or repulsive.

Calculating Gravitational Field Strength

Deriving an expression for
gravitational field strength.

An expression for the gravitational field strength g can be derived by substituting the universal gravitation equation into the simpler expression for field strength shown above.

The value of g is the value for the gravitational field strength at a distance r from the centre of an object with mass m1.

At the surface of a planet like the Earth, the distance r would be the radius of the Earth (6.378 x 106 m) and the field strength at that point would be the same as the acceleration due to gravity of 9.81 ms-2.

Weight

In physics, weight is the name given to the force on an object due to gravity. This force has magnitude often denoted by an italic letter W, which is the product of the mass m of the object and the magnitude of the local gravitational acceleration g. The unit of measurement for weight is that of force, which in the International System of Units (SI) is the newton. For example, an object with a mass of one kilogram has a weight of about 9.8 N on the surface of the Earth, about one-sixth as much on the Moon, and very nearly zero when in deep space far away from all bodies producing gravitational influence.

The acceleration due to gravity
with respect to mass and
distance can be derived by
equating the formula for
gravitational force with the
formula for weight.

The weight force on any object is given by the euqation below:

A person's mass remains constant always and anywhere, but their weight can change because the acceleration due to gravity g can change. Different planets have different values for g at their surface depending on their mass and their radius. The larger their mass, the larger the value of g but the larger the radius, the smaller the value of g. The acceleration due to gravity is directly propotional to the mass of the planet and inversely proportional to the radius squared (shown at right).

You should also note the equivalence of the formulae for gravitational field strength and for acceleration due to gravity. They are exactly the same and have the same units. Any value you calculate for gravitational field strength is the acceleration due to gravity at that point in the field also.

 

Equation - Calculating the Acceleration Due to Gravity

G

m1

r

universal gravitation constant (6.673 x 10-11)

mass of celestial body (e.g. planet)

radius of celestial body

N m2 kg-2

kilograms (kg)

metres (m)

You should use this equation when you are trying to calculate the acceleration due to gravity at the surface of a planet or at an altitude above the planet's surface.

 

The accleration due to gravity at the surface of the Earth is 9.81 ms-2; however, it changes as a result of any of the following:

  • Changes in altitude above the surface of the Earth as a result of mountains or valleys or even satellites in orbit,
  • The distribution of mass in the Earth's crust can vary which means that some areas can be more dense and produce a greater value for g, and
  • The oblation of the Earth makes it 'fatter' at the equator or said another way, it has a greater radius at the equator than it does at the poles.

The acceleration due to gravity and thus the weight of people on other planets or the moon will also vary. Since g varies according to the mass and radius of the planet or celestial body, the weight force experienced will depend on:

  • The mass of the person (object) in the gravitational field,
  • The mass of the planet or body that is the source of the gravitational field, and
  • The radius of the planet or body.

 



Kahn Academy video: Introduction to Gravity. This video covers all of the information covered so far in this section.


Gravitational Potential Energy

Work is defined as the product of force and distance over which the force acts. In gravitational terms, work is done on an object to overcome the gravitational force of attraction when it is raised to an altitude or height. The work done represents an amount of energy GAINED by the object and is equivalent to an increase in gravitational potential energy (GPE).

For an object with mass m being lifted to a certain height h, a force equal and opposite to the weight of the object is needed to lift it. Work is done on the object by a motor or someone's muscles as follows:

The GPE (Ep) is the product of the mass, acceleration due to gravity and the height of the object. While this equation works well when considering the GPE of masses near the surface of the Earth, it does not account for changes in g at large altitudes and is not suitable for considering the GPE around other celestial bodies. Considering GPE in this way also takes the zero point of GPE as the surface of the Earth. This causes problems when considering GPE around other planets and celestial bodies.

Deriving an expression for GPE
through integration of the
equation for the force of
gravity.

We need an expression for GPE that takes the varying magnitude of g into account as well as a definition that has the zero point of GPE at a point in the universe common to all bodies.

GPE is defined as the work done to move an object from infinity to some point within the gravitational field of another object.

In this sense, the zero point of GPE is taken as infinity. The expression for GPE can be obtained by determining the area under a force vs distance graph for an object moving away from the Earth. The area under such a graph is the product of force and distance which you will remember is the work done to move the object against the gravitational attraction of the Earth. We can derive an expression by integrating the equation for the force of gravity with respect to the distance r (at right).

The graph above shows how the negative value
for gravitational potential energy, Ep, increases
with distance up to a maximum value of zero.

When you use the more general form of the gravitational potential energy formula above, the negative sign is significant. At infinity the object has the maximum GPE since it is at the highest 'altitude'. If zero is the maximum value, all other values for GPE must be negative and also represent smaller quantities of GPE. Remember, in the case of GPE, zero is a large quantity and does not mean 'none'.

It should also be noted that GPE is defined by the change in potential energy in moving an object from infinity to a point within the gravitational field of another object. Since the object is losing GPE, a simple calculation of final energy minus initial energy will also yield a negative value. Said another way, the object is doing work rather than having work done on it.

It should also be noted that GPE is proptional to the mass of the object and the negative of GPE is inversely proportional to the radius. The negative sign is significant here because as r increases GPE gets increases but the absolute value of GPE gets smaller. Nevertheless, as r increases, the amount of GPE possessed by an object increases. GPE will always increase with altitude.

Normally in an inversely proportional relationship, as one quanitiy increases, the other decreases. The negative sign though in the GPE formula changes this.

Equation - Calculating Gravitational Potential Energy (GPE)

Ep

G

m1

m2

r

gravitational potential energy

universal gravitation constant (6.673 x 10-11)

mass of heavier object (source of field)

mass of lighter object (in the field)

distance between the centres of each mass

joules (J)

N m2 kg-2

kilograms (kg)

kilograms (kg)

metres (m)

You should use this equation when you are trying to calculate the GPE of an object a certain distance away from the source of the field.

 



A John Rogers video showing how GPE changes with the distance from the Earth.


Comparison of Gravity, Acceleration Due to Gravity and GPE

The graphs on the right show the relationships between a number of variables and distance for a 100 kg space probe in the Earth's gravitational field.

The first graph shows how the force of gravity changes with distance for the space probe. Gravitational force is inversely proportional to the distance squared. This inverse square law causes the force of gravity to drop away very quickly.

By the time the space probe has increased its distance from the centre of the Earth by a factor of 10, the force has dropped to only 9.8 N. At a distance of 20 Earth radii (127 million km from the centre of the Earth) the force is about 2.5 N. At this distance of 127 million km from the Earth's centre, the space probe has only barely passed the orbit of the next planet from Earth, which is Mars. By the time the space probe gets to the orbit of Jupiter, the gravitational force will be only 10 µN and at Saturn's orbit a mere 0.024 µN.

So even though the graph is asymptotic and the space probe will theoretically never escape the pull of the Earth's gravity, the force of gravity is reduced to such small values for these relatively small cosmic distances that it may as well be zero. These small forces would certainly not serve to slow the space probe down to any measurable extent.

The graph for acceleration due to gravity follows the same pattern as that for gravitational force because it also obeys the inverse square law for distance.

The graph for GPE is different in that it has a negative sign such that the distance (and not the distance squared) is inversely proportional to the negative of the magnitude of GPE. This means that as the distance increases, so does GPE! The fact that the GPE is inversely proportional to the distance only means that the graph is not so steep between 1 and 4 Earth radii. This is one of the differences between y=1/x and y=1/x2.

This graph is also asymptotic telling us that the GPE can never actually be zero. As the distance becomes large though, the GPE increases at an increasingly smaller rate until for all intents and purposes, the line is essentially flat and the GPE is constant and very close to zero.