Perpendicular axes theorem (for a planar body): the moment of inertia about an axis perpendicular to the plane equals the sum of the moments of inertia about two mutually perpendicular axes in the plane meeting at the same point: Iz = Ix + Iy. For a ring, Iz (about central axis) = M R^2, and by symmetry Ix = Iy = I(diameter). So M R^2 = 2 I(diameter), giving I(diameter) = M R^2 / 2.

Iz = Ix + Iy

The radius of gyration K is the distance from the axis at which the whole mass of the body can be assumed to be concentrated so as to give the same moment of inertia: I = M K^2, hence K = sqrt(I / M). Its SI unit is the metre.

K = sqrt(I / M)

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

1) Law of orbits: every planet moves in an ellipse with the Sun at one focus. 2) Law of areas: the line joining a planet to the Sun sweeps out equal areas in equal intervals of time (so a planet moves faster when nearer the Sun); this follows from conservation of angular momentum. 3) Law of periods: the square of the orbital period is proportional to the cube of the semi-major axis, T^2 is proportional to a^3.

T^2 = k a^3

For an orbit close to the surface the required centripetal force is provided by gravity, so v_o = sqrt(g R) = sqrt(9.8 * 6.4 x 10^6) = sqrt(6.272 x 10^7) = 7.92 x 10^3 m/s = 7.92 km/s (approximately 8 km/s).

v_o = sqrt(g R)

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.

Orbital velocity near the surface: v_o = sqrt(g R). Escape velocity: v_e = sqrt(2 g R). Dividing, v_e / v_o = sqrt(2 g R) / sqrt(g R) = sqrt(2). Hence v_e = sqrt(2) * v_o (about 1.41 times).

v_e = sqrt(2) * v_o

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

Y = (F L) / (A * delta L). Here F = 10 N, L = 2 m, A = 1 mm^2 = 1 x 10^-6 m^2, delta L = 1 mm = 1 x 10^-3 m. Y = (10 * 2) / (1 x 10^-6 * 1 x 10^-3) = 20 / (1 x 10^-9) = 2 x 10^10 N/m^2.

Y = (F L) / (A * delta L)

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

Viscosity is the internal friction between adjacent layers of a fluid moving with different velocities. Stokes' law: the viscous drag on a small sphere of radius r moving with speed v through a fluid of viscosity eta is F = 6 pi eta r v. Terminal velocity is the constant maximum velocity attained by a body falling through a fluid when the net force (weight minus buoyancy and viscous drag) becomes zero.

F = 6 pi eta r v

Total pressure P = P_atm + rho g h = 1.0 x 10^5 + (1000)(10)(10) = 1.0 x 10^5 + 1.0 x 10^5 = 2.0 x 10^5 Pa.

P = P_atm + rho g h

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

delta L = L alpha (delta T) = 1 * (1.2 x 10^-5) * 100 = 1.2 x 10^-3 m = 1.2 mm.

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

Q = m L = 0.1 kg * 3.36 x 10^5 J/kg = 3.36 x 10^4 J = 33600 J. The temperature stays at 0 degrees C throughout the melting.

Q = m L

Conduction: heat flows through a material from the hotter to the cooler region by molecular vibrations, without bulk movement of the material (e.g. a metal spoon heating up). Convection: heat is carried by the actual movement of the heated fluid (e.g. boiling water, sea breeze). Radiation: heat travels as electromagnetic waves and needs no medium (e.g. heat from the Sun).

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)