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How do you determine the force from a potential energy graph?
1. Find the slope of the potential energy curve at the point of interest. 2. Take the negative of that slope: $$F_{x} = -\frac{dU(x)}{dx}$$.
How do you calculate the total potential energy of a multiple-object system?
Sum the potential energies of all pairs of interacting objects: U_total = U_12 + U_13 + U_23 + ...
How do you calculate the change in potential energy?
The change in potential energy (ΔU) is equal to the negative work done by conservative forces: $$\Delta U = - \int_{a}^{b} \vec{F}_{cf}(r) \cdot d\vec{r}$$
For a potential energy well graph, what points represents stable, unstable, and neutral equilibrium?
Stable equilibrium: Local minima (valleys) | Unstable equilibrium: Local maxima (peaks) | Neutral equilibrium: Flat regions
What are the key differences between conservative and non-conservative forces?
Conservative Forces: Work done is path-independent, net work in a closed loop is zero, potential energy can be defined. Non-Conservative Forces: Work done is path-dependent, net work in a closed loop is non-zero, potential energy cannot be defined.
Compare gravitational potential energy near Earth's surface and for point masses.
Near Earth's Surface: $$Delta U_{g} = mg\Delta y$$, valid for small changes in height. Point Masses: $$U_{g} = -G \frac{m_{1}m_{2}}{r}$$, valid for any separation distance, uses a zero point at infinite separation.
Differentiate between stable and unstable equilibrium points on a potential energy curve.
Stable Equilibrium: Local minimum, system returns to equilibrium after a small disturbance, concave up. Unstable Equilibrium: Local maximum, system moves away from equilibrium after a small disturbance, concave down.