Figure 10.24

1. Recall some examples and discussion from last time—just to refresh your memory (next three slides)…. OSU PH 211, Before Class #29

2. Figure 10.24 OSU PH 211, Before Class #29

3. Figure 10.29 OSU PH 211, Before Class #29

4. Q: In the previous examples, the total mechanical energy (“TE” means Emech) stayed the same no matter what the particle’s position—that is, it’s a level line; TE (Emech) is never increased or decreased, because there is no external work (Wext) being done on the system. Is this generally true in most systems? A: No, there are many instances where an applied force could do work on the system. Moreover, in systems consisting of more than one particle, there is usually a loss of mechanical energy to thermal energy. So the gold TE (Emech) line is not level; it goes “downhill” as the system proceeds from initial position to final position. OSU PH 211, Before Class #29

5. Q: What else can these U vs. x graphs tell us? • A: Potential energy (U) is an “investment” in mechanical energy—where you must do work on the object simply to change its position. Specifically, it’s the work you must do to move the object to a given position from a zero reference point—against some conservative* force Fcons that is acting on the object. • That is: UFcons = ∫(–Fcons)•ds = – ∫(Fcons.s)ds • Examples: UG = –∫(FG)dy = –∫(–mg)dy = mgy • Uspr = –∫(Fspr)dx = –∫(–kx)dx = (1/2)kx2 • *What’s a conservative force? It’s any force whose potential energy investment depends only on the starting and ending positions, not on the path taken. The two conservative forces we have encountered so far are those exerted by gravity and by an ideal spring. Thus, we are confident that any investment of work into gravitational or elastic potential energy can be fully returned (to, say, kinetic energy) without path-related losses to, say, friction or drag. OSU PH 211, Before Class #29

6. The point is, there’s a relationship between a conservative force and the potential energy you invest by doing work against it. And there’s more than one way to express that relationship. We just showed, for instance, how to measure the potential energy, given that we know the conservative force: UFcons = – ∫(Fcons.s)ds But we can go the other way, too—measure the force, given that we know the potential energy: Fcons.s = –dU/ds Examples: FG = –dUG/dy = –mg FSpr = –dUspr/dx = –kx So even when we don’t know the nature of the conservative force, if we have its potential energy graph, we can see how it is acting by looking at the (negative of the) slope of that potential energy graph. OSU PH 211, Before Class #29

7. A particle moves along the x-axis with the potential energy shown. The conservative x-force acting on it when it is located at x = 4 m is… • 4 N. • 2 N. • 1 N. • –1 N. • E. –2 N. OSU PH 211, Before Class #29

8. A particle moves along the x-axis with the potential energy shown. The conservative x-force acting on it when it is located at x = 4 m is… • 4 N. • 2 N. • 1 N. • –1 N. • E. –2 N. OSU PH 211, Before Class #29