Design Terminology e = Strain δl = change in length of a body, mm l = original length of a body, mm $e = {δl}/l $ σ = stress, N/$mm^2$ e = strain E = modulus of elasticity, N/$mm^2$ σ α e σ = Ee $E = σ/e$
The centre of mass of n Point Masses Moment = (Mass) . (Lever arm) $\ov{x} = {∑↙{i=1}↖n m_i x_i}/{∑↙{i=1}↖n m_i}$ If mass distribution is continuous $\ov{x} = {∫ x dm}/{∫ dm}$ C.G. of an arc of a curve $\ov{x} = {∫ x dm}/{∫ dm} = {∫ x ρ ds}/{∫ ρ ds}$ ; $\ov{y} = {∫ y dm}/{∫ dm} = {∫ y ρ ds}/{∫ ρ ds}$ if ρ is constant, we have: $\ov{x} = {∫ x ds}/{∫ ds}$ ; $\ov{y} = {∫ y ds}/{∫ ds}$ Note: 1. For y = f(x) replace ds = $√{1 + ({dy}/{dx})^2} dx$ 2. For x ...
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Representation of area as a double integral Type I: Area enclosed by plane curves expressed in Cartesian co-ordinates Area = $∫↙{a}↖{b} dx ∫↙{f_1(x)}↖{f_2(x)} dy$ Area = $∫↙{c}↖{d} dy ∫↙{g_1(x)}↖{g_2(x)} dx$ Note: 1. The area A included by curve y = f(x), the x-axis, and the ordinates x = a and x = b is given by Area = $∫↙{a}↖{b} y dx = ∫↙{a}↖{b} f(x) dx$ 2. The area A included by the curve x = f(y), the y-axis and the abscissa y = c and y = d is given by A = $∫↙{c}↖{d} x dy = ∫↙{c}↖{d} f(y) dy$ ...
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Double integration Representation of area as a double integral: $A = lim↙{δx→0} ∑↙{x=a}↖{x=b} y . δx$ Expressed in integral notation as: $A = ∫↙a↖b y dx$ or $∫↙a↖b f(x) dx$ $∬↙R$ dx dy represents the area of the region R $∬↙R$ f(x, y) dx dy represents some physical quantity related to the area of region R. Properties of the double integral 1. $∬↙R$ k f(x, y) dA = k $∬↙R$ f(x, y) dA k is constant, free from x and y 2. $∬↙R$ [f(x, y) ± g(x, y)] dA = $∬↙R$ f(x, y) dA ± $∬↙R$ g(x, y) dA 3. ...
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Cone with vertex at the origin $ax^2 + b y^2 + c z^2 + 2 fyz + 2 gzx + 2hxy = 0 $ which is a homogenous equation. Conversely, every homogenous equation of second degree in x, y, z represents a cone whose vertex is at the origin Cor. : If the line $x/l = y/m = z/n$ is a generator of the cone (whose vertex is at the origin) $ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0$, then direction cosines (or direction ratios) l, m, n satisfy the equation of cone $al^2 + ...
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Equations of sphere in different forms (A) Centre and radius form $(x – a)^2 + (y – b)^2 + (z-c)^2 = r^2$ (B) General form $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$ Centre: (-u, -v, -w) Radius: $r=√{u^2 + v^2 + w^2 – d}$ (C)Intercept form: To find the equation of the sphere which cuts off intercepts a, b, c from ox, oy, oz axes: $x^2 + y^2 + z^2 – ax – by – cz = 0$ (D) Diameter form: To find the equation of the sphere described on the join ...
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Relations between three coordinate systems Relations between cartesian and spherical polar system of coordinates are x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ Relations between cartesian and cylindrical system of coordinates are x = ρ cosϕ y = ρ sinϕ z = z Note: (i) In cartesian coordinate system: -∞ < x < ∞, -∞ < y < ∞, -∞ < z < ∞ (ii) In spherical polar system: 0 < r < ∞, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π (iii) In cylinderical coordinate system: 0 < ρ < ∞, ...
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Rectification of plane curve when its equation is in polar form r=f(θ) 1. r = f(θ) $s = ∫↙{θ_1}↖{θ_2} √{r^2 + ({dr}/{dθ})^2} dθ$ 2. θ = f(r) $s = ∫↙{r_1}↖{r_2} √{1 + r^2 ({dθ}/{dr})^2} dθ$ 1. Cartesian curves $\table,\text"Equation of curve",\text"Formula in differential calculus", \text"Formula in integral calculus";1,y=f(x),ds = √{1 + ({dy}/{dx})^2} dx, s = ∫↙{x_1}↖{x_2} √{1 + ({dy}/{dx})^2} dx;2,x=g(y), ds = √{1 + ({dx}/{dy})^2} dy, s = ∫↙{y_1}↖{y_2} √{1 + ({dx}/{dy})^2} dy; 3,x = f_1 (t)\, y=f_2 (t), ds = √{({dx}/{dt})^2 + ({dy}/{dt})^2} dt, s = ∫↙{t_1}↖{t_2} √{({dx}/{dt})^2 + ({dy}/{dt})^2} dt$ 2. Polar Curves: $\table ,Equation of curve,Formula in ...
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Rule I: Integral with limits (a, b) as constants If I(α) = $∫↙{a}↖b f(x, α) dx, then {dI}/{dα} = ∫↙{a}↖b {δ}/{δα} f(x, α) dx$ Rule II: Integral with limits as functions of parameter: Leibnitz’s Rule ${dI}/{dα} = d/{dα}∫↙{a(α)}↖{b(α)} f(x, α) dx = ∫↙{a(α)}↖{b(α)} δ/{δα} f(x, α) dx + f(b, α) {db}/{dα} -f (a,α) {da}/{dα}$ Error function 1. $erf(x) = 2/{√π} ∫↙{0}↖x e^{-u^2} du$ 2. Complementary error function $erfc(x) = 2/{√π} ∫↙{x}↖∞ e^{-u^2} du $ 3. Alternate definition of error function $erf(x) = 1/{√π} ∫↙{0}↖{x^2} e^{-t} t^{- 1/2} dt$ Properties of error functions (1) erf(∞) = 1 (2) erf(0) = 0 ...
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Reduction formulae for sinusoidal functions 1. To find reduction formula for $∫ sin^n x dx$, n is positive integer ≥ 2 and evaluate completely $∫↙{0}↖{π/2} sin^n x dx$ $I_n = {n-1}/{n} I_{n-2}$ $\table ∫↙{0}↖{π/2} sin^n x dx,=,{n-1}/n {n-3}/{n-2} …. 3/4 1/2 π/2\; \text"if n is even";,=,{n-1}/n {n-3}/{n-2}…..4/5 2/3 1 \; \text"if n is odd"$ $\table ∫↙{0}↖{π/2} cos^n x dx,=,{n-1}/n {n-3}/{n-2} …. 3/4 1/2 π/2\; \text"if n is even";,=,{n-1}/n {n-3}/{n-2}…..4/5 2/3 1 \; \text"if n is odd"$ Additional Results: I. $∫↙{0}↖{π} sin^n x dx$ = $2 ∫↙{0}↖{π} sin^n x dx$ , for all n integral values of n II. $\table ∫↙{0}↖{π} ...
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