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where dh is change of height
We calculate the total work by definite integral
h 2
A   mgdh  mgh  mgh . (5.20)
1
2
h 1
According to the formula (5.18) the sign of work is plus, because

  (pay attention to fig.4.7) .But the same work as follows from
2
(5.20), is with minus sign. What to do? No problem, rewrite the equation
(5.20) in the next form (5.21) , where sign of work corresponds to fact.

A   (mgh  mgh 1 ). (5.21)
2
Thereby, work done by gravity force doesn’t depend on
trajectory and only depends on final and initial position of a body in

gravity field. The value of such work is defined by the difference of
function which has the following form and is called the potential energy
of a body in the gravity field.

E  mgh. (5.22)
P
According to (5.21)
A    E . (5.23)
P
The gravity force is example of conservative force
A conservative force is a force with the property that the work done

in moving a particle between two points is independent of the path
taken. Equivalently, if a particle travels in a closed loop, the net work
done (the sum of the force acting along the path multiplied by the
distance travelled) by a conservative force is zero.

A conservative force is dependent only on the position of the object. If a
force is conservative, it is possible to assign a numerical value for the

potential at any point. When an object moves from one location to
another, the force changes the potential energy of the object by an
amount that does not depend on the path taken. If the force is not
conservative, then defining a scalar potential is not possible, because

taking different paths would lead to conflicting potential differences
between the start and end points. A non conservative force is a force that
acts on a particle (or point),in such way that the work done by this force

in moving this particle from one point to another is dependent on the
path taken. To put it other words , the work done depends on the path
itself. For example, a friction force is non conservative because the work
done by friction always acts in the direction of travel and therefore

depends on the length.

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