For a ring about its central axis, I = M R^2 = 2 * (0.5)^2 = 2 * 0.25 = 0.5 kg m^2.

I = M R^2

A geostationary satellite appears stationary relative to the Earth because it revolves in step with the Earth's rotation. Conditions: (i) its orbital period must equal 24 hours (equal to Earth's rotation period); (ii) it must orbit in the equatorial plane, in the same (west-to-east) sense as the Earth, at a height of about 36000 km.

Hooke's law: within the elastic limit, the stress developed in a body is directly proportional to the strain produced, i.e. stress / strain = a constant (the modulus of elasticity). The elastic limit is the maximum stress up to which a body returns to its original shape and size on removing the deforming force; beyond it, permanent (plastic) deformation occurs.

stress = E * strain

Pascal's law: a pressure applied to an enclosed incompressible fluid is transmitted equally and undiminished to every part of the fluid and to the walls of the container. Application: the hydraulic lift/brake, where a small force on a small piston produces a large force on a large piston because pressure is the same throughout.

F1 / A1 = F2 / A2

For the steady flow of an incompressible fluid, the product of area of cross-section and speed is constant along the tube: A1 v1 = A2 v2 (A v = constant). It is a statement of conservation of mass - where a pipe is narrow the fluid flows faster, and where it is wide it flows slower.

A1 v1 = A2 v2

Surface tension is the property by which the free surface of a liquid behaves like a stretched elastic membrane; it is the force per unit length acting along the surface (SI unit: N/m). Example: small water droplets and mercury beads become spherical (minimum surface area), and some insects can walk on water.

T = F / L

The coefficient of linear expansion (alpha) is the fractional increase in length per degree rise in temperature: alpha = (delta L) / (L * delta T). The increase in length is delta L = L alpha (delta T), and the new length is L' = L (1 + alpha delta T). SI unit of alpha: per kelvin (K^-1).

delta L = L alpha delta T

Specific heat capacity is the heat required to raise the temperature of 1 kg of a substance by 1 K (Q = m c delta T; unit J/kg/K) - the temperature changes. Latent heat is the heat required to change the state of 1 kg of a substance at constant temperature (Q = m L; unit J/kg) - the temperature stays constant during the phase change.

Q = m c delta T ; Q = m L

Newton's law of cooling states that the rate of loss of heat of a body is directly proportional to the difference in temperature between the body and its surroundings, provided this difference is small: (- dQ/dt) is proportional to (T - T_surroundings).

dQ/dt = -k (T - T0)

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.

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.)

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

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

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)

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.

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

U = - G M m / r. It is negative because the gravitational force is attractive: zero potential energy is taken at infinity, and as the mass is brought closer, work is done by gravity, lowering the energy below zero. The negative sign shows the mass is in a bound state.

U = - G M m / r

The coefficient of restitution (e) is the ratio of the relative velocity of separation after collision to the relative velocity of approach before collision: e = (velocity of separation) / (velocity of approach). For a perfectly elastic collision e = 1; for a perfectly inelastic collision e = 0.

e = (v2 - v1) / (u1 - u2)