This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt: ... work done on a system of bodies acted an by conservative forces, in moving from one configuration P to a second configuration Q, is independent of the series of configurations through which the system moves inpassing from P to Q. To prove this, let us denote the work done in passing from P to Q through one series of ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt: ... work done on a system of bodies acted an by conservative forces, in moving from one configuration P to a second configuration Q, is independent of the series of configurations through which the system moves inpassing from P to Q. To prove this, let us denote the work done in passing from P to Q through one series of configurations by IVi that done hi passing through any second series by W2, and that done in returning from Q to P by any third series of configurations by W3. If we pass from P to Q by the first series and back from Q to P by the third, the total work done is nil, so that wt+wt = o. So, also, if we pass from P to Q by series 2 and back from Q to P by series 3, Wt + Wt = 0. Thus W1 = JF2, which proves the theorem. 132. Definition. Taking any configuration P as standard, the work done in moving a system of bodies from the configuration P to the configuration Q is spoken of as the potential energy of configuration Q. The potential energy, accordingly, measures the work which has been stored up in placing the system in configuration Q. Theorem. The work done in moving a system from a configuration (1) to a second configuration (2) against conservative forces is W2--Wv where Wv W2 are respectively the potential energies in configurations (1) and 2). For if P is the standard configuration, the work from P to (1) is W1; the work from P to (1) plus that from (1) to (2) is W2, so that the work from (1) to (2) is W2-W1. 133. Theorem. If a system of bodies is in a configuration of potential energy W, and if x, y, z are the coordinates of any particle, the resultant force acting on the particle has components dW dW dW dx cy cz To prove this, let us imagine that we give the system a small displacement, which consists in moving the single particle at x, y, ...
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