Power and potential curves
Power is the rate of delivering energy, and a potential energy curve is a map of where a particle can and cannot go.
Power
Power is work per unit time, measured in watts where 1 W = 1 J/s. For motion at speed v driven by force F along the motion, power is also F times v, which is why a car climbing a steep grade at constant speed needs the same power no matter which gear it uses. Doubling the climb speed doubles the power demanded.
P = \frac{W}{t} = FvReading U(x) curves
A potential energy graph encodes the force: F = -dU/dx, so the force points downhill on the curve and vanishes at the flat spots. A particle with total energy E can only be where U(x) is at most E, and the turning points are the x values where U(x) = E, since there the kinetic energy hits zero and the particle turns back.
F = -\frac{dU}{dx}, \quad K = E - U(x)Equilibrium
The flat spots of a U(x) curve are equilibrium points. At a valley bottom U has a local minimum: displace the object either way and the force pushes it back toward center, like a marble in a bowl. At a hilltop U has a local maximum: any nudge and the force pushes it further away, like a pencil on its tip. Try U(x) = 3x^2 J, whose minimum sits at x = 0: the force is F = -6x, so at x = 1 m it points back with 6 N, confirming stable equilibrium.
\text{stable: } U \text{ min}, \quad \text{unstable: } U \text{ max}Flashcard
Power is energy delivered per second
Check your understanding
A spring with k = 200 N/m is compressed 0.30 m and holds a 0.50 kg block against it at rest. When released, the block leaves the spring at the spring's natural length. What is the block's speed at x = 0.15 m from natural length? Take the total energy as the initial spring energy.