The zeroth law states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. It leads to the concept of temperature - a property that is the same for all bodies in thermal equilibrium.

At constant volume all the heat supplied goes to increase the internal energy (no work is done). At constant pressure the gas also expands and does external work, so extra heat is needed for the same temperature rise; hence Cp > Cv. The relation (Mayer's relation) is Cp - Cv = R, where R is the universal gas constant.

Cp - Cv = R

Kelvin-Planck statement: it is impossible to construct an engine that, working in a cycle, converts all the heat absorbed from a source completely into work with no other effect. (Equivalently, Clausius statement: heat cannot flow of its own accord from a colder body to a hotter body.)

For an isothermal process, W = n R T ln(V2/V1) = 1 * 8.31 * 300 * ln(2) = 8.31 * 300 * 0.693 = 1727 J (approximately 1.73 kJ).

W = n R T ln(V2/V1)

From kinetic theory, P = (1/3) rho <v^2>, where rho is the density and <v^2> is the mean-square speed of the molecules. Since the kinetic energy per unit volume is (1/2) rho <v^2>, we get P = (2/3) * (kinetic energy per unit volume). Thus pressure is two-thirds of the translational KE per unit volume.

P = (1/3) rho <v^2>

Law of equipartition: in thermal equilibrium, the total energy is shared equally among all degrees of freedom, each contributing (1/2) k T of energy per molecule. A monatomic molecule has 3 degrees of freedom (translational only); a diatomic molecule at ordinary temperatures has 5 (3 translational + 2 rotational).

Energy per degree of freedom = (1/2) k T

Average translational KE per molecule = (3/2) k T = (3/2)(1.38 x 10^-23)(300) = 1.5 * 1.38 x 10^-23 * 300 = 6.21 x 10^-21 J.

KE = (3/2) k T

Simple harmonic motion is an oscillation in which the restoring force (or acceleration) is directly proportional to the displacement from the mean position and is always directed towards it. Defining equation: a = - omega^2 x, where omega is the angular frequency and x the displacement.

a = - omega^2 x

For SHM of amplitude A, kinetic energy KE = (1/2) m omega^2 (A^2 - x^2) and potential energy PE = (1/2) m omega^2 x^2. Adding, total energy E = KE + PE = (1/2) m omega^2 A^2. This is constant - it does not depend on x - and equals (1/2) m omega^2 A^2. Energy shuttles between KE (maximum at the mean position) and PE (maximum at the extremes).

E = (1/2) m omega^2 A^2

T = 2 pi sqrt(m / k). Since T is proportional to sqrt(m), quadrupling the mass (m -> 4m) makes sqrt(4m) = 2 sqrt(m), so the time period doubles.

T = 2 pi sqrt(m / k)

Angular frequency omega = 2 pi / T = 2 pi / 2 = pi rad/s. Maximum velocity v_max = A omega = 0.05 * pi = 0.05 * 3.14 = 0.157 m/s (about 15.7 cm/s).

v_max = A omega

Forced oscillations occur when a body is made to oscillate under an external periodic force with the frequency of that force. Resonance is the special case when the driving frequency equals the body's natural frequency; the amplitude of oscillation then becomes maximum. Example: a child's swing pushed at its natural frequency.

In one time period T, a wave advances by one wavelength lambda. So wave speed v = distance / time = lambda / T. Since frequency f = 1/T, we get v = f lambda. This holds for all progressive waves.

v = f lambda

A progressive wave transfers energy continuously in one direction, and every particle has the same amplitude but oscillates with a phase lag. A stationary wave is formed by two identical waves travelling in opposite directions; it does not transfer net energy, has fixed nodes (zero amplitude) and antinodes (maximum amplitude), and the amplitude varies from point to point.

For a string of length L, linear density mu and tension T, the fundamental frequency (first harmonic) is f1 = (1/2L) sqrt(T/mu), because the fundamental mode fits half a wavelength in the length L. Harmonics (overtones) are the higher allowed frequencies, which are integer multiples of the fundamental: fn = n f1 (n = 1, 2, 3, ...).

fn = (n / 2L) sqrt(T / mu)

Beats are the periodic rise and fall in the loudness of sound produced when two waves of slightly different frequencies superpose. The beat frequency equals the difference of the two frequencies: 260 - 256 = 4 beats per second.

f_beat = |f1 - f2|

For a source approaching a stationary observer, f' = f * v / (v - v_s) = 340 * 340 / (340 - 34) = 340 * 340 / 306 = 377.8 Hz (about 378 Hz). The pitch appears higher.

f' = f v / (v - v_s)

Relative velocity of a body A with respect to B is the velocity of A as seen from B: v_AB = v_A - v_B. For the trains, v = 60 - 40 = 20 km/h, so the faster train appears to move ahead at 20 km/h relative to the slower one.

v_AB = v_A - v_B

The restoring force at extension x' is F = k x'. The work done in stretching the spring from 0 to x is W = integral of (k x') dx' from 0 to x = (1/2) k x^2. This work is stored as elastic potential energy: U = (1/2) k x^2.

U = (1/2) k x^2

A rolling body has both translation and rotation. Its total kinetic energy is KE = (1/2) m v^2 + (1/2) I omega^2, where the first term is the translational KE of the centre of mass and the second is the rotational KE about the centre. Using v = R omega and I = m K^2, this becomes KE = (1/2) m v^2 (1 + K^2/R^2).

KE = (1/2) m v^2 + (1/2) I omega^2