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4.8. Relationship between potential energy and power

Lecture



Each point of the potential field corresponds, on the one hand, to a certain value of the force vector   4.8.  Relationship between potential energy and power acting on the body and, on the other hand, some value of potential energy   4.8.  Relationship between potential energy and power . Consequently, there must be a definite connection between force and potential energy.

  4.8.  Relationship between potential energy and power

To establish this connection, we calculate the elementary work   4.8.  Relationship between potential energy and power performed by field forces with small displacement   4.8.  Relationship between potential energy and power body occurring along an arbitrarily chosen direction in space, which we denote by   4.8.  Relationship between potential energy and power . This job is equal to

  4.8.  Relationship between potential energy and power

Where   4.8.  Relationship between potential energy and power - projection of force   4.8.  Relationship between potential energy and power on direction   4.8.  Relationship between potential energy and power .

Since in this case the work is done at the expense of the stock of potential energy   4.8.  Relationship between potential energy and power , it is equal to the loss of potential energy   4.8.  Relationship between potential energy and power on the axis segment   4.8.  Relationship between potential energy and power :

  4.8.  Relationship between potential energy and power

From the last two expressions we get

  4.8.  Relationship between potential energy and power

From where

  4.8.  Relationship between potential energy and power

The last expression gives the average   4.8.  Relationship between potential energy and power on the segment   4.8.  Relationship between potential energy and power . To

get value   4.8.  Relationship between potential energy and power at the point you need to make the limit:

  4.8.  Relationship between potential energy and power

Because   4.8.  Relationship between potential energy and power may change not only when moving along the axis   4.8.  Relationship between potential energy and power but also when moving along other directions, the limit in this formula represents with robots the so-called partial derivative of   4.8.  Relationship between potential energy and power by   4.8.  Relationship between potential energy and power :

  4.8.  Relationship between potential energy and power

This relation is valid for any direction in space, in particular, for the directions of Cartesian coordinate axes x, y, z:

  4.8.  Relationship between potential energy and power

This formula defines the projections of the force vector on the coordinate axes. If these projections are known, the force vector itself is determined:

  4.8.  Relationship between potential energy and power

in math vector   4.8.  Relationship between potential energy and power ,

where a is the scalar function x, y, z, is called the gradient of this scalar denoted by   4.8.  Relationship between potential energy and power . Hence the force is equal to the gradient of potential energy taken with the opposite sign.

  4.8.  Relationship between potential energy and power (4.15)

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Physical foundations of mechanics

Terms: Physical foundations of mechanics