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)

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

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)

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

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

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).

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

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>

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

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

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

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)

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

Uses: (i) to check the dimensional consistency of an equation; (ii) to convert a quantity from one system of units to another; (iii) to derive the relation between physical quantities. Limitations: (i) it cannot find dimensionless constants (like 1/2 or 2 pi); (ii) it cannot be used if a quantity depends on more than three others, or on trigonometric/exponential/logarithmic functions.