Portland State University

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1 page - 16 Formal definition of POTENTIAL ENERGY (valid for conservative forces only) Given one type of conservative force F Definition of 'Potential energy difference" UB - UA= - = - conserv

2 page - 17 conserv conserv For

3 PH-212 A. La Rosa Portland State University Potential Energy and Conservative Forces For a given conservative force F, a scalar potential energy U is associated to such a force. Vector U (1) Scalar Such an association is built through the definition of the potential energy difference UB - UA as equal to negative value of the work W done by the conservative force F conserv (when taking the particle from an initial potion A to a final position B), UB - UA - W (A (2) B) Alternative notation: By making the point B a general position x, and the point A a reference fixed position, then one obtains the potential energy U as a function of x, UX - Uref point - W (Ref. point X) (3) Or, U(x) = Uref point - W (Ref. point X) (We will see below why it is convenient to put the minus sign in the definition above) (4)

4 Examples In the case of the spring force, for example, we found in PH211 that, W (A B) = - (1/2) k [ (xb)2 - (xa)2 ] (5) xa xb 0 Hence, the definition (2), UB - UA - W (A B), implies UB - UA + (1/2) k [ ( xb)2 - (xa)2 ] If we take Point A as the origin and as reference point as well (xa=0), and make xb a general point of coordinate x, then we obtain, U(x) = U(0) + (1/2) k x2 Arbitrarily we assign, U(0) = 0 which makes (6) to become, (6)

5 U(x) = (1/2) k x2 In short (7) U (8) = - kx U(x) = (1/2) k x 2 Similarly, in the case of the gravitational force. Previously, in PH-211, we found, W (A B) = - mg [ zb - za ] (9) a value that is independent of the particular trajectory followed by the object during its motion from A to B. zb B g za ma Accordingly, the definition (2), UB - UA - W (A B), implies UB - UA + mg ( zb - za ) This quantity depends only on the initial and final positions of the particle (10)

6 If we take as a reference point A a point of coordinate z0, and make B a general point of coordinate z, then we obtain, U(z) = U(0) + mg ( z - zo ) (11) Arbitrarily we assign, U(0) = U0 which makes (11) to become, U(z) = U0 + mg ( z - zo ) In short (12) U (13) = - mg z U(z) = U0 + mg ( z - zo ) In the definition of the potential energy difference established by a conservative force, UB - UA - W (A B), the negative sign in front of W is quite convenient for later expressing the conservation of the mechanical energy. In effect, as we know, from the work / kinetic-energy theorem we have W (A B) = KB - KA Combining this result with the definition UB - UA - W (A B)

7 leads to KB + UB = KA + UA = const [ Notice, had we chosen VB (14) - VA + W (A B) we would have ended up with something like KB - VB = KA - VA K + U =E ] is defined as the mechanical energy of the particle Since A and B are two arbitrary points along the trajectory of the particle, the result in the previous page indicates that the mechanical energy conserves thought the motion Alternative definition of the Potential Energy In the previous section we defined a potential difference UB - UA in terms of the work done by the conservative force F, UB - UA zb -W (A B). We offer here an alternative, but equivalent, definition in terms of an external force. B m Conservative force A The graph above shows the force lines of a conservative force. Let s assume that the particle of mass m moves from A to B along the indicated trajectory (in blue color). As we know, the

8 particle will pick up some kinetic energy as well as some potential energy change along the way. Question: What does an external agent will have to do in order to move the particle from A to B, along the same trajectory, but at constant velocity (velocity ~ 0)? Motion at constant velocity implies equilibrium. It means that the external force will have to be of the same magnitude of the conservative force, but pointing in opposite direction. That is, F external (15) z zb B Fext F Conservative Fext m force X A Fext = - F Then B) UB - UA - W (A F B) = Wext (A F ext

9 That is UB - UA B) Wext (A F ext (16) Work done by the external force to take the particle from A to B at constant speed The definitions (2) and (16) should provide the potential-energy difference between the points A and B.