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

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.

Roads are banked at curves so that the horizontal component of the normal reaction provides the centripetal force needed to turn, reducing reliance on friction and the risk of skidding. On a frictionless banked road, the ideal (safe) speed is v = sqrt(r g tan(theta)).

v = sqrt(r g tan(theta))

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)

Young's modulus (Y) measures resistance to change in length (longitudinal stress/strain). Bulk modulus (B) measures resistance to change in volume (volume stress/strain) under uniform pressure. Shear (rigidity) modulus (G) measures resistance to change in shape (shearing stress/strain).

v_t = (2 r^2 (rho - sigma) g) / (9 eta), where r is the radius of the sphere, rho its density, sigma the density of the fluid, g the acceleration due to gravity and eta the coefficient of viscosity. It is obtained by balancing the weight against the buoyant force and the viscous (Stokes) drag.

v_t = 2 r^2 (rho - sigma) g / (9 eta)

Stefan's (Stefan-Boltzmann) law: the total energy radiated per unit area per unit time by a black body is proportional to the fourth power of its absolute temperature, E = sigma T^4. Wien's displacement law: the wavelength at which the emission is maximum is inversely proportional to the absolute temperature, lambda_max * T = constant (= 2.9 x 10^-3 m K).

E = sigma T^4 ; lambda_max T = b

A refrigerator is a heat engine working in reverse: it uses external work W to extract heat Q_c from a cold body and reject a larger heat Q_h to the hot surroundings, so Q_h = Q_c + W. Its coefficient of performance is beta = Q_c / W = Q_c / (Q_h - Q_c). A good refrigerator has a high beta.

beta = Q_c / (Q_h - Q_c)

The mean free path is the average distance a gas molecule travels between two successive collisions. It increases when the number density of molecules is low and when the molecular diameter is small; that is, it is inversely proportional to the number density and to the square of the molecular diameter.

lambda = 1 / (sqrt(2) pi d^2 n)

Velocity v = dx/dt = A omega cos(omega t), with magnitude v = omega sqrt(A^2 - x^2); it is maximum (A omega) at the mean position (x = 0) and zero at the extremes. Acceleration a = dv/dt = - A omega^2 sin(omega t) = - omega^2 x; its magnitude is maximum (A omega^2) at the extreme positions (x = +/- A) and zero at the mean position.

v = omega sqrt(A^2 - x^2) ; a = - omega^2 x

Newton assumed sound propagation in a gas is isothermal, giving v = sqrt(P / rho), where P is pressure and rho density; this gave a value about 15% lower than the measured value. Laplace corrected this by treating the process as adiabatic (compressions and rarefactions are too rapid for heat exchange), giving v = sqrt(gamma P / rho), where gamma is the ratio of specific heats; this matches experiment.

v = sqrt(gamma P / rho)

The centre of mass is the point at which the entire mass of the system can be considered to be concentrated, and where an applied external force produces the same acceleration as on the whole system. For two particles of masses m1 and m2 at positions x1 and x2, x_cm = (m1 x1 + m2 x2) / (m1 + m2).

x_cm = (m1 x1 + m2 x2) / (m1 + m2)

With depth d, g decreases as g' = g (1 - d/R), assuming uniform density, because only the mass within the smaller inner sphere attracts the body. At the centre (d = R) the value becomes zero, since the mass is symmetrically distributed all around.

g' = g (1 - d/R)

When the lift accelerates upward, the apparent weight (normal reaction) is N = m (g + a) = 50 (10 + 2) = 50 * 12 = 600 N. (At rest it would be m g = 500 N, so the person feels heavier.)

N = m (g + a)

At the highest point, gravity alone must provide the centripetal force, so m g = m v_top^2 / r, giving the minimum speed at the top v_top = sqrt(g r). Using energy conservation between the lowest and highest points, the minimum speed at the bottom is v_bottom = sqrt(5 g r).

v_top = sqrt(g r) ; v_bottom = sqrt(5 g r)

Resolution of a vector is the process of splitting it into two or more components, usually along mutually perpendicular directions. For a vector A at angle theta to the x-axis, the rectangular components are Ax = A cos(theta) along the x-axis and Ay = A sin(theta) along the y-axis, with A = sqrt(Ax^2 + Ay^2).

Ax = A cos(theta) ; Ay = A sin(theta)

A quasi-static process is one carried out infinitely slowly so that the system stays in thermal and mechanical equilibrium with its surroundings at every stage. It is important because only for such (reversible) processes can the state variables (P, V, T) be defined throughout, and the work done can be represented as an area on a P-V diagram.

In an open pipe (open at both ends) both even and odd harmonics are present; the fundamental is f1 = v/(2L) and the harmonics are f1, 2 f1, 3 f1, ... In a closed pipe (closed at one end) only odd harmonics are present; the fundamental is f1 = v/(4L) and the harmonics are f1, 3 f1, 5 f1, ... So an open pipe of the same length gives a higher fundamental and a richer set of harmonics.

Open: fn = n v/2L ; Closed: fn = (2n-1) v/4L

Distance in the nth second: s_n = u + (a/2)(2n - 1). With u = 0, a = 2, n = 5: s_5 = 0 + (2/2)(2*5 - 1) = 1 * 9 = 9 m.

s_n = u + (a/2)(2n - 1)

Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, directed along the line joining them: F = G m1 m2 / r^2, where G is the universal gravitational constant (6.67 x 10^-11 N m^2 / kg^2).

F = G m1 m2 / r^2