Unizor - Physics4Teens - Energy - Energy of Gravitational Field - Gravitational Field

Notes to a video lecture on http://www.unizor.com

Gravitational Field

Studying forces, we have paid attention to a force of attraction, that exists between any material objects, the force of gravity.
For example, if a comet from outer space flies not far from a Sun, it is attracted by Sun and changes its straight line trajectory.

In Mechanics we used to see the force as something between the objects touching each other, like a man pushing a wagon. In case of gravity the force obviously exists, but it acts on a distance, in "empty" space.
In Physics this concept of force acting on a distance is described by a term field. Basically, field is the area in space where some force acts on all objects or only on objects that have specific property. The force in this case depends on a point in space and an object that experiences this force and, as a result of the action of force, changes its movement.

Gravitational field exists around any material object (the source object of a field) and acts as an attraction towards this source object, experienced by any other material object (probe object) positioned in this field.
As described in the "Gravity, Weight" chapter of "Mechanics" part of this course, the magnitude of the gravitational force F is proportional to a product of masses of a source object and a probe object, M and m, and it is inversely proportional to a square of a distance r between these objects:
F = G·M·m /r²
where G - a constant of proportionality, since the units of force (N - newtons) have been defined already, and we want to measure the gravitational force in the same units as any other force.

The direction of the gravitational force acting on a probe object is towards the source object.

Let's return to our example of a comet flying not far from the Sun and, being attracted to the Sun, changing its trajectory. Obviously, to change the trajectory, some energy must be spent. So, we conclude that gravitational field has certain amount of energy at each point that it spends by applying the force onto a probe object.

To quantify this, assume that the source of gravity is a point mass M fixed at the origin of coordinates. Position a probe object of mass m at coordinates {r₁,0,0} and let it go. The force of gravity will cause the motion of this probe object towards the center of gravity, the origin of coordinates, so the movement will be along the X-axis. Let the ending position of the probe object be {r₂,0,0}, where r₂ is smaller then r₁. Let x be a variable X-coordinate (distance to the origin).

According to the Universal Law of Gravitation, the force of attraction of a probe object towards the source of a gravitational field at distance x from the origin equals to
F = −G·M·m /x²
where minus in front of it signifies that this force is directed opposite to increasing the X-coordinate.
This force causes the motion and, therefore, does some work, moving a probe object from point {r₁,0,0} to point {r₂,0,0} along the X-axis. To calculate the work done by this variable force, we can integrate F·dx from x=r₁ to x=r₂:
W_[r1,r2] = ∫_[r1,r2]F·dx =
= −∫_[r1,r2]G·M·m·dx /x² =
= G·M·m /x|_[r1,r2] =
= G·M·m /r₂ − G·M·m /r₁ =
= (G·M /r₂ − G·M /r₁)·m

The expression
V(r) = −G·M/r
is called gravitational potential.
It's a characteristic of a gravitational field sourced by a point mass M at a distance r from a source.
It equals to work needed by external forces to bring a probe object of mass m=1 to a point at distance r from a source of the field from infinity.
Indeed, set m=1, r₁=∞ and r₁=r in the above formula for work W_[r1,r2] and take into consideration that gravitational field "helps" external forces to move a probe object, so the external forces spend negative amount of energy.

Using this concept of gravitational potential V(r), we can state that, to move a probe object of a unit mass from distance r₁ relative to a source of gravitational field to a distance r₂ relative to its source in the gravitational field with gravitational potential V(r), we have to spend the amount of energy equal to V(r₁)−V(r₂).
For a probe object of any mass m this amount should be multiplied by m.
If r₂ is greater than r₁, that is we move a probe object further from the source of gravity, working against the gravitational force, this expression is positive, we have to apply effort against the force of gravity. In an opposite case, when r₂ is smaller than r₁, that is we move closer to a source of gravity, the gravitational force "helps" us, we don't have to apply any efforts, and our work is negative.

Therefore, an expression E_P=m·V(r) represents potential energy of a probe object of mass m at a distance r from a source of a gravitational field with gravitational potential V(r).

A useful consequence from a concept of a gravitational potential is that the force of gravity can be expressed as the derivative of the gravitational potential.
F = G·M·m /r² = m·dV(r)/dr
which emphasizes the statement that the gravitational potential is a characteristic of a field itself, not its source.
We, therefore, can discuss gravitational field as an abstract concept defined only by the function called gravitational potential.