5.4. Kinetic Energy

Kinetic energy W_k is the energy a body possessed by virtue of its motion. Any body in motion can set other bodies in motion by colliding with them. An action of the force F causes acceleration a of the material point, namely changing its velocity v. Changes of v mean changes of the kinetic energy. Hence,

dW_k = A = F· dS· cos.

Figure 5.3

The total alteration of the kinetic energy W_k along the way 12 is given by:

.

Because a·cos = a_ = dv/dt, dS = v·dt we have:

Therefore

From the condition W_k = 0 if v = 0 it follows that const = 0. Hence,

The kinetic energy is expressed in joules. The kinetic energy depends only on velocity v (W_k = f(v)).

5.5. Potential Energy

The energy that a body (or a system of bodies) has by virtue of its position or configuration is called potential energy W_p.

The potential energy depends only on coordinates of material points included in a system (W_p = f(x, y, z)).

The reduction of the potential energy during the transition of a system from the state 1 to the state 2 is measured by work produced by all potential forces (internal or external) that act on a system, namely:

W_p(1) - W_p(2) = A_1-2,

or for infinitesimals

A = - dW_p.

The potential energy depends only on the initial and the final positions of the points and does not depend on the way of a transition.

If the external potential forces are non-stationary then the potential energy of the system depends not only on configuration of a system but on time t also (W_p = f(x, y, z, t)). Besides, these forces produce the work only when a displacement of a system takes place. That’s why in the general case:

Let us consider the stationary case of the system consisting of one material point on which the potential force F acts:

Hence,

F_x = - ∂W_p/∂x, F_y = - ∂W_p/∂y, F _z= - ∂W_p/∂z

Using a definition of the operator of gradient

it is possible to obtain:

5.6. Total Energy of System. Conservation of Energy

The theorem about alteration of the kinetic energy of a system:

An alteration of the kinetic energy dW_k for a system is equal to the work done by the internal and the external forces applied to the material points of a system:

.

Here A is the work done by the internal and the external forces applied to the i-th material point

A_i = A_i^pf + A_i^npf,

where A_i^pf is the work of the potential forces and A_i^npf is the work of the non-potential forces.

From the definition of the potential energy (see 5.5) it follows:

Then,

It is possible to transform the last expression and receive the law of the energy alteration of a system:

Here the first term describes the non-stationary origin of the external potential fields and the second term is the total work of all non-potential forces.

The mechanical energy (or the total mechanical energy) is the sum of the mechanics motion energy W_k and the energy of interactions W_p:

W = W_k + W_p.

The system is called conservative if all external potential forces are stationary (W_p/t = 0) and all external and internal non-potential forces do not produce the work ( ). In this case:

dW = d(W_k + W_p) = 0 and W = const.

The law of energy conservation for conservative system.

The mechanical energy of conservative system does not change with time.