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Online text: openstax.org/books/university-physics-volume-2/pages/7-2-electric-potential-and-potential-difference

Cited passages: University Physics Volume 2, §7.2, equations 7.4–7.7: energy per charge, work and potential difference.

Cited passages: OpenStax, University Physics II, §7.2: potential difference, work, and energy.

Online text: openstax.org/books/university-physics-volume-2/pages/1-5-phase-changes

Cited passages: University Physics Volume 2, §1.5, figure 1.18 and equations 1.7–1.8: molecular attractions and constant-temperature phase conversion.

Cited passages: Ling, Moebs and Sanny, OpenStax, Phase changes: fixed-pressure plateaux and latent heat; OpenStax, Phase changes and heating curves; OpenStax, §1.5: fusion and specific latent heat.

Online text: openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy

Cited passages: University Physics Volume 2, §3.2, “Internal Energy”, equation 3.5 and the discussion of molecular motion; §3.2, “Internal Energy”: molecular speeds relative to the center of mass; moving-train and roof examples.

Cited passages: OpenStax, §3.2: pressure-volume work and the isothermal integral.

Online text: openstax.org/books/university-physics-volume-1/pages/11-4-precession-of-a-gyroscope

Cited passages: OpenStax University Physics Volume 1, section 11.4: Precession of a Gyroscope; University Physics Volume 1, section 11.4, equation 11.12 and the torque and precession-circle derivation.

Online text: openstax.org/books/university-physics-volume-1/pages/16-2-mathematics-of-waves

Online text: openstax.org/books/university-physics-volume-1/pages/11-2-angular-momentum

Online text: openstax.org/books/university-physics-volume-1/pages/15-5-damped-oscillations

Cited passages: University Physics Volume 1, §15.5, equations 15.23–15.26 and the force discussion preceding 15.23.

Online text: openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy

Cited passages: §13.4, “Circular Orbits” and “Energy in Circular Orbits”: centripetal balance and negative total energy.

Cited passages: OpenStax: circular speed.

Online text: openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy

Cited passages: §13.3, equations 13.4–13.6 and “Escape velocity”: infinity reference, small test mass and exterior launch; §13.3, “Energy and gravitationally bound objects”: scalar energy and finite-body collisions.

Cited passages: OpenStax: escape as the zero-energy threshold; OpenStax: work against gravitational attraction.

Online text: openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia

Cited passages: Moebs, Ling and Sanny, University Physics Volume 1, section 10.5, Parallel-Axis Theorem: centroidal reference axis and squared axis separation; Moebs, Ling and Sanny, University Physics Volume 1, section 10.5: continuous distributions, uniform discs and compound objects.

Online text: openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy

Cited passages: William Moebs, Samuel J. Ling and Jeff Sanny, University Physics Volume 1, section 10.4: fixed-axis rotation and the sum of particle kinetic energies.

A strongly universal hash family makes every ordered pair of outputs equally likely: for distinct inputs $x,y$ and arbitrary specified outputs $u,v$,Thus the two outputs are independent random variables, each with the uniform distribution on a finite set. Adding the $q$ equal ordered pairs gives collision probability $q/q^2=1/q$, so the family is also a universal hash family.

Here is a completely enumerable example. In the finite field ${\mathbb{F}}_5$, take the affine functions $h_{a,b}(x)=ax+b$, using modular arithmetic modulo $5$. Choose $a$ and $b$ as independent random variables, each with the uniform distribution on a finite set of all five finite field elements $0,1,2,3,4$, including $a=0$. There are $25$ equally likely hash functions. For $x=0,y=1$, the ordered pair is $(b,a+b)$. For instance $(a,b)=(2,3)$ gives $(h(0),h(1))=(3,0)$; $(0,3)$ gives $(3,3)$. Every prescribed $(u,v)$ occurs exactly once, at $b=u,a=v-u$.

Each row contains five choices, so each individual output has probability $5/25=1/5$. Each cell has probability $1/25$, and the five diagonal cells give collision probability $5/25=1/5$. For any distinct $x,y$, the same conclusion follows from $a=(v-u)/(y-x)$ and $b=u-ax$: a nonzero finite field element has a unique inverse. This is the beauty of the construction: two small random coefficients provide exact pair balance without choosing five unrelated values.

The example uses equal-sized input and output sets to make every choice visible; it illustrates the family definitions rather than compression. In a larger input set, a hash function cannot avoid all collisions into five buckets. What a universal hash family controls is the fraction of its members that collide on each fixed pair, rather than the existence of any collision anywhere.

Restricting to $a\ne 0$ leaves $20$ permutations with zero collision probability for distinct inputs. Individual outputs remain uniform, but given $h(0)=u$, the four values other than $u$ each have conditional probability $1/4$ and $u$ has zero. Hence these outputs are not independent random variables: this is a universal hash family but not a strongly universal hash family. Finally, the full $25$-member example contains five constant functions, with immediately visible collisions. Its pair guarantee concerns a random choice of hash function, and makes no claim of cryptographic collision resistance.

Online text: web.physics.ucsb.edu/~lecturedemonstrations/Composer/Pages/52.33.html

Cited passages: UCSB Physics lecture demonstration 52.33, opening apparatus description: crank-turn counter, frictional work and thermistor readout.

Cited passages: Opening apparatus paragraphs: aluminum cylinder, wrapped rope, suspended mass, crank, thermistor, resistance conversion and turn counter; Operating the demonstration, initial resistance/temperature calibration and steady cranking sequence; final minimum thermistor resistance; Final operating paragraphs: starting below room temperature and ending above it to approximately balance room heat exchange; omitted accessory capacities and rough estimates.