By J.R. Calvert

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Yn-2)] (h/3) x ("first+ last+ fourxodds + twoxevens") Runge-Kutta Method: Second Order: Fourth Order: Yn+l = Yn +~{i(x,,y,)+ f(x, +h,y, +k1)} 2 1 Yn+l =y, +-(kl +2k2 +2k3 +k4 ) 6 kJ = hf(x,,y,) k2 =hf(x, k3 = hf( x, k4 +~,y, + 'i) +~,y, + k;) = hf(x, +h,y, +k 3 ) 3-22 4. 1 Analysis of Experimental Data Probability Distributions for Discrete Random Variables Notation: P(r) = f(r) =>the probability distribution of random variable r isf(r) n mean value of r f-l = L:rJ(r;) i=l n (J2 variance of r (J standard deviation of r (~) binomial coefficient = L:r/ f(r;)- ,U 2 i=l ( n ) n!

3 - - Moments of Inertia Definitions: Moment oflnertia: I xx = Jr dm 2 M r is the perpendicular distance of the element of mass dm from the axis XX; the integral is taken over the whole mass of the body. Radius of gyration k is defined by I= Mk 2, for a specified axis. Parallel axis theorem: Ixx= IGG + My02 M is the mass of the body, y 0 is the perpendicular distance between an axis GG through the centre of mass and another axis XX parallel to GG. IGG is the minimum moment of inertia for all axes in the given direction.

No addition or subtraction): (s; +( m ; + .. etc where n, m etc are the powers of x, y etc inf( ). Notes: (a) The maximum possible error ( S z = 0 z S x + 0 z SY + ... etc) is rarely ax oy of interest in engineering. ±ox may be treated as a normally (b) Instrument 'rounding off error distributed error by the equivalence Sx ~ ~&. 4-4 5. 1 Second Moment ofArea Definition: lxx = JJ(y- Yt)2dxdy The double integral is taken over the whole area of the shape. Similar expressions apply to axes in other directions.