The definition of work could thus be restated as the amount of energy transferred by forces

The tractor raises the weight over the pulley, increasing its gravitational potential energy.

In example (1 2 Tractor IP), the increase in the height of the weight, , is the same as the distance the tractor travels, call it d. For simplicity, we discuss the case where the tractor raises the weight at constant speed, so that there is no change in the kinetic energy of the weight, and we assume that there is negligible friction in the pulley, so that the force the tractor applies to the rope is the same as the rope’s upward force on the weight.

The increase in the weight’s potential energy is given by (), so the work done by the tractor on the weight equals , the product of the force and the distance moved: , or

(2.1) dW=F dx or .

The definition of work could thus be restated as the amount of energy transferred by forces.

When a contact force acts between two objects and the two surfaces do not slip past each other, the work done equals Fd, where d is the distance traveled by the point of contact.

If the force is constant and along the same line as the motion:

(2.2) .

One of the most important types of motion occurs when the force F is a function of the coordinate x alone. The integral is the work done by the force when the weight goes from to x.

We now define the potential energy as the work done by the force when the weight goes from to some chosen standard point x:

(2.3)

The work done by a force F equals the area under the curve on a graph of F versus x.

In general case of three dimensions (2.3) should be written in a vector form:

(2.4) .

The most important example is the work done by gravity when the change in height is not small enough to assume a constant force. Newton’s law of gravity is

,

which can be integrated to give

(2.6)

The techniques for calculating work can also be applied to the calculation of potential energy. If a certain force depends only on the distance between the two participating objects, then the energy released by changing the distance between them is defined as the potential energy, and the amount of potential energy lost equals minus the work done by the force,

D U = – W.