PHYS233 Combined

Newton’s 3rd Law (force pairs)

The force that object 1 exerts on object 2 is equal in magnitude and opposite in direction to the force that object 2 exerts on object 1, even if their charges/masses differ. Different masses only change acceleration, not the interaction force.

Coulomb force symmetry

For two charges separated by r, each experiences the same |F| = k|q1 q2|/r^2 (opposite directions). Bigger charge does NOT mean it “feels” a bigger force—both feel the same magnitude.

Repulsion vs attraction (like charges)

Like charges repel: the force on each charge points away from the other charge along the line connecting them.

Force diagram sanity check (2 charges)

In any correct force diagram for two interacting objects: forces must be equal length (equal magnitude) and opposite direction, acting on different objects.

When can forces be unequal?

Never for an interaction pair (Newton 3rd). You can have unequal NET forces on two objects if they have additional forces (friction, normal, etc.), but the mutual pair stays equal/opposite.

Acceleration vs force

Acceleration depends on net force AND mass: a = Fnet/m. Same force does not mean same acceleration if masses differ.

Same force for same time → impulse rule

If the same constant force acts for the same time interval Δt, the impulse J = FΔt is the same, so the momentum change Δp is the same.

Impulse definition

J = ∫F dt. For constant force, J = FΔt. Impulse equals momentum change: J = Δp.

Momentum definition

p = mv. Momentum depends linearly on mass and velocity.

Same Δp does NOT mean same v

If Δp is the same but masses differ, the smaller mass must have larger Δv (because Δv = Δp/m).

Same Δp does NOT mean same KE

KE = (1/2)mv^2 = p^2/(2m). If p is the same, the smaller mass has larger KE (because KE ∝ 1/m for fixed p).

Same force, same displacement → same work

If a force does the same work over a displacement, both objects receive the same ΔK (work–energy theorem), regardless of mass.

Work–energy theorem

Net work done on an object equals the change in its kinetic energy: Wnet = ΔK.

Work definition (dot product)

Work by a force is W = F·Δr = FΔr cosθ. Only the component of force parallel to motion does work.

Positive work meaning

If the force has a component in the direction of motion (cosθ > 0), the force does positive work and tends to increase kinetic energy.

Negative work meaning

If the force has a component opposite the motion (cosθ < 0), the force does negative work and tends to decrease kinetic energy.

Zero work meaning

A force does zero work if it is perpendicular to displacement (cosθ = 0) or if there is no displacement.

Tension in uniform circular motion

In uniform circular motion, tension (or centripetal force) points radially inward while displacement is tangential, so W = 0 over any segment; over a full circle, still W = 0.

Normal force work on level surface

For motion along a flat surface, the normal force is perpendicular to displacement, so it does zero work.

Gravity work depends on vertical change

Work by gravity near Earth: Wg = mgΔy (positive if moving downward, negative if upward). It depends on Δy and m (not on path shape).

Gravity work for different masses

Dropping two masses from the same height: the heavier mass has larger |Wg| because Wg = mgh.

Spring force direction

A spring force points opposite displacement from equilibrium: Fspring = −kx.

Spring potential energy

Uspring = (1/2)kx^2. More compression/stretch stores more energy.

Conservative force definition

A force is conservative if work depends only on start and end positions (path independent). Gravity and ideal spring forces are conservative.

Conservative round trip rule

For a conservative force, total work over a closed path is zero. Example: compress spring then return to rest → net spring work over the round trip is 0.

Nonconservative force definition

A force is nonconservative if its work depends on path length/details. Kinetic friction is nonconservative.

Friction work sign

Kinetic friction typically does negative work because it points opposite the direction of motion.

Applied force vs friction net work

You can do positive work pulling an object while friction does negative work; net work depends on which magnitude wins.

Same 10 J twice ≠ same speed jump

Because KE ∝ v^2, equal work increments do not produce equal speed increments. Going 0→2 m/s and 2→4 m/s do NOT generally require the same work.

Speed doubling energy rule

Doubling speed requires quadrupling kinetic energy: (1/2)m(2v)^2 = 4*(1/2)mv^2.

KE vs v relationship

Kinetic energy increases with the square of speed (v^2), so high-speed changes are energetically expensive.

Momentum vs v relationship

Momentum increases linearly with speed (v), so it scales “less aggressively” than kinetic energy.

Same KE does NOT mean same momentum

At fixed KE, p = √(2mK). Larger mass gives larger momentum for the same kinetic energy.

Same momentum does NOT mean same KE

At fixed p, K = p^2/(2m). Smaller mass gives larger kinetic energy.

Work by constant force over distance

For constant F along motion: W = FΔx. Bigger distance or bigger force → more work.

Power meaning

Power is the rate of doing work: P = W/Δt. High power means fast energy transfer.

Energy conservation (mechanical)

If only conservative forces act, mechanical energy E = K + U stays constant.

When mechanical energy changes

If nonconservative forces (like friction) do work, mechanical energy changes (usually decreases) though total energy (including thermal) is conserved.

Choosing the “system” in energy problems

Conservation statements depend on what you include in the system. If you include Earth + object, gravity becomes internal; if not, gravity is external work.

Potential energy graph: force link

In 1D, Fx = −dU/dx. Force points toward decreasing U.

Stable equilibrium on U(x)

A stable equilibrium is at a local minimum of U(x): small displacements create restoring force back toward the minimum.

Unstable equilibrium on U(x)

An unstable equilibrium is at a local maximum of U(x): small displacements push the object away.

Turning point meaning

A turning point occurs when K = 0, so total energy equals potential energy: E = U.

Matter current (flow rate) definition

Volumetric flow rate Q is volume per time crossing a surface: Q = Av.

Continuity (incompressible)

In steady incompressible flow, Q is constant along the pipe: A1v1 = A2v2.

Speed in narrow section

If cross-sectional area decreases, speed increases (since Q constant). Narrower pipe → larger v.

Flow rate same everywhere (steady)

In steady incompressible flow, the volumetric flow rate is the same through wide and narrow sections (what goes in must come out).

Mass flow rate form

Mass flow rate I = ρAv. For incompressible liquids, ρ is constant.

Pressure drop direction in viscous flow

In viscous flow, pressure decreases in the direction of flow; the pressure difference drives flow against viscous drag.

Viscous drag in pipes concept

Fluid “sticks” to pipe walls; layers shear; friction between layers produces viscous resistance that opposes flow.

Hagen–Poiseuille big idea

For laminar flow in a cylindrical tube: ΔP = ZQ, with Z ∝ (μL)/(R^4). Small radius changes dominate.

Radius scaling trap (Poiseuille)

If radius doubles (R→2R), resistance drops by 16×, so flow rate increases by 16× for the same ΔP (NOT 4×).

Length scaling trap (Poiseuille)

If length doubles (L→2L), resistance doubles, so flow rate halves for the same ΔP.

Viscosity scaling

If viscosity μ increases, resistance increases proportionally and flow rate decreases for the same ΔP.

Series resistance (pipes)

Pipes in series add resistances: Ztotal = Z1 + Z2 + …

Parallel resistance (pipes)

Pipes in parallel reduce resistance: 1/Zeq = 1/Z1 + 1/Z2 + … so Zeq is smaller than each individual resistance.

Parallel intuition (why smaller)

Parallel gives more “pathways” for flow, increasing total flow for the same ΔP, equivalent to lower overall resistance.

Hydrostatic pressure change

Pressure increases with depth: ΔP = ρgh (deeper → higher pressure).

Buoyant force definition (Archimedes)

Buoyant force equals the weight of displaced fluid: Fb = ρfluid Vdisplaced g.

Floating condition (equilibrium)

For a floating object at rest: Fb = weight = mg, so ρfluid Vdisp g = ρobject Vobject g.

Fraction submerged rule

fraction submerged = ρobject / ρfluid (for floating objects).

Sinks vs floats comparison

If ρobject > ρfluid → sinks. If ρobject < ρfluid → floats. If equal → neutrally buoyant (fully submerged with no net force).

Heat vs temperature difference

Temperature is a measure related to average microscopic energy per degree of freedom; heat is energy transferred because of temperature difference.

Direction of heat transfer (net)

When two objects at different temperatures touch in isolation, net heat flows from hotter to colder until equilibrium.

Microscopic “two-way” exchange

Even while net heat flows hot→cold, microscopic energy transfer events can occur both directions; “only hot→cold” is too strict.

Thermal equilibrium condition

At equilibrium, temperatures are equal; there is no net heat flow.

Energy conservation in thermal contact

In an isolated two-object system, Qhot + Qcold = 0 (energy lost by one equals energy gained by the other).

Final temperature mixing formula concept

Tf is a weighted average by heat capacities: bigger mc means it “resists” temperature change more and pulls Tf toward its initial T.

Specific heat meaning

High specific heat means it takes more heat to change temperature: Q = mcΔT.

Heat capacity meaning

C = mc, so Q = CΔT. Bigger C means smaller ΔT for the same Q.

Entropy big idea

Entropy measures number of microscopic arrangements (microstates) compatible with a macrostate: S = kB lnΩ.

Second law (isolated system)

For an isolated system evolving spontaneously, total entropy increases or stays the same: ΔStotal ≥ 0.

Entropy and “direction” of spontaneity

Processes tend to move toward macrostates with more microstates (more probable), which typically means higher total entropy.

Entropy of two blocks in contact

When two blocks exchange energy in isolation, total entropy increases until equilibrium (maximum Stotal).

Total entropy constant in real spontaneous contact?

No: if heat flows irreversibly, Stotal increases. Constant Stotal requires an ideal reversible process (not typical for real contact).

Reversible process concept

A reversible process proceeds through equilibrium states and can be reversed by an infinitesimal change with no net entropy production.

Entropy constant in isolated process

For an isolated system, ΔS = 0 implies the process is reversible (ideal) or nothing changes; it does NOT automatically mean “not reversible.”

Fundamental assumption of statistical mechanics

All microstates consistent with a given macrostate are equally probable (the “equal a priori probability” assumption).

Equilibrium = most probable macrostate

Equilibrium corresponds to the macrostate with the largest multiplicity (largest Ω), i.e., the one most likely to occur.

Multiplicity (Ω) meaning

Ω counts the number of microstates; larger Ω means higher probability for that macrostate.

Combinatorics: distributing identical items

The number of ways to distribute q identical objects among N containers is Ω = (q+N−1)! / [q!(N−1)!].

Apple/bucket type question logic

Identical items + distinct buckets → use “stars and bars” (the Ω formula). The count is combinations, not permutations.

Marbles/blocks type question logic

Independent choices multiply: (choices for marble)×(choices for block). If 5 marbles and 4 blocks → 20 combos.

Temperature (statistical definition)

Temperature relates to how entropy changes with energy: 1/T ∝ dS/dE (higher T means adding energy increases S more slowly).

What equalizes at thermal equilibrium

It is temperature that equalizes between systems in contact, not necessarily energy, not necessarily number of quanta, and not necessarily entropy.

Energy quanta sharing (different oscillator counts)

If two Einstein solids have different numbers of oscillators, they generally will NOT end with equal numbers of quanta; equilibrium is equal temperature (matching slopes of S vs q).

“Hotter” with same energy analogy

Two systems can have the same total energy but different temperatures if their numbers of available microstates (degrees of freedom/pockets/oscillators) differ.

Degrees of freedom and thermal energy

More available modes (“pockets”) means the same energy is spread thinner, often leading to lower temperature.

Equipartition (classical)

Average energy per quadratic degree of freedom is (1/2)kBT; total average energy depends on number of active degrees of freedom.

Why vibrational modes may be “off”

At room temperature, some vibrational quanta may be too high in energy to be populated, so they don’t contribute to heat capacity.

Heat capacity and degrees of freedom

Higher active degrees of freedom → higher heat capacity because energy can be stored in more modes.

Spontaneous vs fast

Second law predicts direction (toward higher entropy) but not the rate; a process can be spontaneous yet slow.

“Only way reversible is entropy increases” trap

False: reversible processes have no entropy production; for isolated systems, reversible means ΔS = 0, not ΔS > 0.

Hot-to-cold: “only direction heat can flow” trap

False if interpreted microscopically (energy can transfer both ways). True statement is about NET heat flow.

Hot-to-cold: “net heat flow” statement

True for isolated objects in contact: net heat goes from higher T to lower T until equilibrium.

Entropy change when heat flows

If heat flows irreversibly from hot to cold, total entropy increases (ΔStotal > 0).

Entropy sign clue for T/F

If an isolated system does something “spontaneous,” expect ΔStotal ≥ 0; if a claim says ΔStotal < 0 without outside input, it’s wrong.

Quick T/F test: same force for same time

Conclude same impulse → same Δp, but NOT same v, NOT same KE, unless masses same.

Quick T/F test: same force for same displacement

Conclude same work → same ΔK, but NOT same speed, NOT same momentum, unless masses same.

Quick T/F test: same starting rest + same work

Because Kfinal = W (if starting at rest and only that force does work), equal work means equal final kinetic energy.