PHYC10003 Chap.3 Potential Energy and Particle Systems
Potential Energy and Particle Systems
Potential-energy curves encode force direction in their slope and kinetic energy in their separation from the total-energy level. This chapter distinguishes stable equilibrium, turning points and allowed regions, then expands the analysis from one particle to a system. A boundary makes internal force pairs cancel in the momentum balance while external impulse changes total momentum.
Force points down the local potential gradient. A minimum is stable because nearby forces point back toward it, while a maximum is unstable because nearby forces point away. A horizontal total-energy line creates a second reading: the vertical gap above the potential is kinetic energy, intersections are candidate turning points, and regions above the total are forbidden.
Zero speed at an intersection does not establish equilibrium, because the slope can still give nonzero force. System analysis begins with a drawn boundary. Interactions between included particles are internal and cancel in total momentum as paired vectors, even while they accelerate individual particles. Forces or impulses crossing the boundary change total momentum and centre-of-mass motion.
Isolation alone does not guarantee constant kinetic energy. A spring release can convert stored energy into motion, and a sticking collision can convert kinetic energy into internal forms while momentum remains conserved. Naming the boundary, external-impulse approximation, and relevant energy store keeps these conclusions distinct and prevents one conservation law from being used as another.
What this chapter covers
- 01
Force as negative potential gradient
- 02
Stable and unstable equilibrium
- 03
Allowed motion and turning points
- 04
Kinetic energy from curve separation
- 05
Internal and external interactions
- 06
System momentum
Worked example · free
Use isolation to solve a spring release
- 1Choose both carts and the spring as one system.
- 1Set initial and final total momentum equal to zero.
- 1Write 2.0 times minus 3.0 plus 3.0 times v equal to zero.
- 1Solve v as 2.0 m/s to the right.
Key terms
- Potential Gradient
- A potential gradient is the spatial rate at which potential energy changes.
- Stable Equilibrium
- Stable equilibrium occurs where small displacements produce restoring forces toward the equilibrium point.
- Turning Point
- A turning point is a position where kinetic energy is zero and motion reverses in the allowed region.
- Internal Force
- An internal force is exerted between objects that both lie inside the chosen system boundary.
- External Impulse
- External impulse is momentum transferred to a system by forces exerted from outside its boundary.
Potential Energy and Particle Systems FAQ
Does low potential energy always mean large force?
No. Force depends on the slope of potential energy, not its absolute value. A deep flat minimum can have zero force at its centre, while a higher but steep region can have a large force. Read height, slope, and curvature as separate graph features before drawing a physical conclusion.
Is every point with zero speed an equilibrium?
No. Zero speed describes one instant. Equilibrium requires zero net force. A turning point can have zero speed while a nonzero force immediately reverses the motion.
Why do internal forces cancel for system momentum?
Internal forces occur in equal and opposite pairs between members of the same system. Their vector contributions cancel in the total momentum balance, although they can change individual momenta and redistribute energy.
Exam move
Annotate potential graphs in two colours: one for local slope and force direction, another for the vertical gap between total and potential energy. At several marked positions, state slope sign, force sign, allowed or forbidden motion, relative speed, and whether the point is an equilibrium or turning point.
Keep those questions separate because curve height alone does not determine force and zero speed alone does not determine equilibrium. Practise classifying a local minimum, local maximum, flat interval, and ordinary turning point by describing what happens after a small displacement. Add a horizontal total-energy line and move it upward or downward to see how allowed regions merge or disappear.
Convert selected energy gaps into speeds only after verifying that kinetic energy is nonnegative. Then reverse the process: given a force-direction diagram, sketch a compatible qualitative potential curve by remembering that force points down the potential gradient. For particle-system questions, draw a boundary first and label every interaction as internal or external before invoking momentum conservation.
Redraw the same event with a different boundary and identify which forces change category. Use vector arrows for internal action-reaction pairs and confirm that they cancel in the total momentum balance without claiming that each particle has zero acceleration. Separate momentum conservation from kinetic-energy conservation by comparing a spring release, an elastic collision, and a sticking collision.
In each case, name the energy store or dissipation that explains the kinetic-energy change. Use centre-of-mass motion as a check: with negligible external impulse, the centre-of-mass velocity remains fixed even when individual velocities change sharply. End each answer by naming the boundary, the external impulse assumption, and whether the conclusion concerns total momentum, kinetic energy, or both.
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