Specific Heat of a Substance Q = m . C . Δt Q = heat transfer in kJ, m = mass in kg C = specific heat Δt = $t_2 – t_1$ = temperature change in Kelvin (final temperature – initial temperature) Unit: kJ/kg K Enthalpy H = U + p . V H = specific enthalpy U = specific internal energy of gas First Law of Thermodynamics (Joule’s Experiment) First law as applied to closed system ∮d’ Q = ∮d’ w First law as applied to closed system processes Q = W + ΔU Coefficient of Performance (C.O.P) $W ...
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Resistance Welding (Electric Resistance Welding) $H = I^2 RT$ Some terms, H = heat generated in joules, I = current in amperes, R = resistance at the contacting area between two metal parts in ohms, t = time of current flow in seconds
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 $E = σ/e$
Belt Drives Some terms: Consider a belt drive, d = diameter of the driving pulley, mm D = diameter of the driven pulley, mm n = speed of the driving pulley, r.p.m. N = speed of the driven pulley, r.p.m. t = thickness of the belt, mm V = linear speed of the belt, mm/s i = speed ratio or velocity ratio Linear speed of the belt, $V = {πdn}/{60}$ or $V = {πDN}/{60}$ $n/N = D/d$ Speed ratio, or velocity ratio, $i = {\text"Speed of driving pulley"}/{\text"Speed of driven pulley"}$ $i = n/N = D/d$ if thickness of belt ...
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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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