Mechanical work

If you accelerate a body to a greater speed by applying a force to the body, you increase the kinetic energy of the body. Similarly, if you decelerate the body to a lesser speed by applying a force, you decrease the kinetic energy of the body. When we consider these changes in kinetic energy we say that you have transferred energy to the body or the body has transferred energy to you.

In such a transfer of energy via force, work W is said to be done on the body by the force. More formally, we define work as follows:

Work W is energy transferred to or from a body by means of a force
acting on the body. Energy transferred to the body is positive work,
and energy transferred from the body is negative work

"Work," then, is transferred energy; "doing work" is the act of transferring the energy. Work has the same units as energy and is a scalar quantity.

Let us now relate the work done on a body by force and the corresponding change in the kinetic energy of the body. If the force changes the speed of the body, it also changes the kinetic energy of it. If the kinetic energy is the only type of the body energy being changed by the force, then the change of kinetic energy is equal to the work W done by the force:

Here is the initial kinetic energy and is the kinetic energy after the work — the energy transfer — is done.

The right-hand of equality can also be written

The last two equations are equivalent statements of the work-kinetic energy theorem.

If the body's energy other than kinetic energy is changed by the force,
then the work-kinetic energy theorem does not apply

Now we would like to refer the change of the body kinetic energy to the magnitude F of the force causing the change. We begin with a particle; the only type of energy this simplest type of body can have is kinetic energy. In Fig. 3.4, a particle moves along x axis on a horizontal frictionless surface while a constant force acts on it at a constant angle to the particle's path. Because the horizontal force component F cos gives the particle an acceleration a_x along the path, the force changes the particle's velocity from its initial value Thus the force also changes the particle's kinetic energy.

Suppose the force acts on the particle through a displacement giving it a velocity which magnitude is

where

Multiplying both sides of this equation by the particle mass m and rearranging we obtain:

Here the left side is the difference between the particle initial kinetic energy and the final kinetic energy This difference is the change of the particle kinetic energy due to the force Substituting F cos for the product ma_x according to Newton's second law, we get

= F×s· cos

We can see that the right side of equation gives the work W done on the particle by force Thus, we may write

W = F·s· cos

If the angle is less than 90°, then the work W is positive, which means that energy is transferred to the particle and the kinetic energy of the particle increases. If is greater than 90° (up to 180°), then the work W is negative, which means that energy is transferred fromthe particle and the kinetic energy of the particle decreases.

The last equation demonstrates that in addition to joule, another SI work unit is the newton - meter(N × m).

The right side of equation of the work W is equivalent to the scalar (or dot) product So in vector form the expression for W is

This equation is especially useful when and are given in the unit-vector notation.

Дата добавления: 2015-11-16