Work and power

Assume that a material point under constant force on indefinitely small time interval dt is moved by a distance d (Fig. 2.7). The scalar product of the force vector in the displacement vector is called the elementary work force :

(2.42)

Where the angle between the vectors and d .If angle then the work that occurs, when moving of the point is positive.

If the angle when the force is perpendicular to the displacement then and the work is not performed. If the angle then work will be negative.

Write the vectors and d through the projections on the coordinate axes:

Where projections of the vector on the coordinate axes, x, y, z – coordinates of the point at which the force is applied .

Then the formula (2.42) for the elementary work looks:

To find the work A of the force in the case of a finite displacement from point 1 to point 2, conditionally divide it into infinitesimal displacement d , each of which force can be considered constant and the elementary work defined with (2.42). Then the work of force in case of moving from point 1 to point 2.

Taking into account the relation (2.43), the expression for the work can be written as:

Thus, the work of a variable force on the final movement of the AB is determined

line integral along the trajectory.

The unit of work in the CI is the joule, which is, according to (2.42) work of 1 H force on the way to 1 m:

1Jl = 1H

In some cases, it is important to know not only the work, but also the time during which work is done. The work that is done per unit of time is called power:

Power is equal to the scalar product of vectors of force and velocity.

Formula (2.46) can be written in terms of the projections on the coordinate axes:

From (2.46) followed that the power can be increased by increasing both the force and velocity.

If the power is given as a function of time, the work can be determined for a period of time from to . From (2.46) follows that:

The CI unit of power is Watt: 1 W - is the power at which 1s. the work of 1 Jl is performed.