MATHEMATICAL FORMULA:Force: force is defined as an external agent which changes or tends to change the state of rest or of uniform motion of a body. Force is a vector quantity.Characteristics of a force: the quantities by which a force is fully represented are called characteristics of the force. The various characteristics of a force are1. Magnitude: the quantity of a force is called its magnitude. 2. Direction: the line along which the force acts is called direction or line of action of

the force. It is represented by (a) indicating the angle that the line of action of the force makes with the fixed or reference axes such as OX, OY or with any other horizontal or vertical line. (b) Indicating the angle that the line of action of the force makes with the geographical axes such as North, South, East or West.

3. Sense: the way in which the force acts along its line of action is known as sense or nature of the force, i.e. push or pull.

4. Point of application: the point at or through which the force acts on a body is called its point of application.

Equilibrium: any system of forces which keeps the body at rest is said to be in equilibrium, or when the condition of the body is unaffected even though a number of forces acted upon it, it is said to be in equilibrium.Laws of Equilibrium: there are two laws of equilibrium depending upon the forces or the couple acting on a rigid body.

1. Force law or equilibrium2. Moment law of equilibrium.

Force law of Equilibrium: for any system of forces keeping a body in equilibrium, the algebraic sum of forces, in any direction is zero, i.e. Σ F = 0.This law is applicable to all systems of coplanar forces and includes. (a)Two force principle: if a body in equilibrium is acted upon by two forces, they must be equal, opposite and collinear.(b) Three force principle: if a body in equilibrium is acted upon by three forces, then the resultant of any two forces must be equal, opposite and collinear with the third force. (c) Four force principle: if a body in equilibrium is acted upon by four forces, then the resultant of any two forces must be equal, opposite and collinear with the resultant of other two forces.Moment Law of Equilibrium: For any system of forces keeping a body in equilibrium, the algebraic sum of the moments of all the forces about any point in their plane is zero i.e. Σ M =0 or Σ F.d = 0. This law is applicable only to coplanar, non-concurrent (parallel or non-parallel) force system.

Lami’s theorem: If three coplanar forces acting on a point in a body keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces. Thus, P/ sin α = Q / sin β = R / sin γ.

Trusses: a truss is an articulated structure composed of straight members arranged and connected in such a way that they transmit primarily axial forces.

1. Plane truss: if all the members lie in one plane, the truss is called plane truss. 2. Space truss: a three dimensional truss is called a space truss.

Assumptions: 1. Each truss is composed of rigid members all lying in one plane. 2. Weight of the members is neglected because they are small in comparison with the

loads.3. Forces are transmitted from one member to another through smooth pins fitted in

the members.Work done:

1. Work done by an elastic string during stretching = (initial tension +final tension)/2 .Extension. = Mean tension. Extension.

2. Work done in resisting a body by a height h = mgh.3. Work done in rotation = Torque. Angle turned through.4. Work done by a couple = Moment of the couple. Angle in radians turned through.

Energy: energy is the capacity to do work. Infact, energy is possessed by a body, whereas work is done by a force.Different forms of Energy: Mechanical energy, Electrical energy, Heat energy, Light energy, Sound energy, Pressure energy etc.Mechanical energy: The energy possessed by a body due to its position or motion is called mechanical energy. Mechanical energy is of two kinds.

1. Kinetic energy: the energy possessed by a body by virtue of its motion is called Kinetic energy. It is measured by the amount of work which the body can do due to its motion. Kinetic energy of translation: the energy due to rectilinear motion of a body is called kinetic energy of translation. KE = mv2 / 2. Kinetic energy of rotation: the kinetic energy due to rotary motion of the body is called kinetic energy of rotation. KE of Rotation = I ω2 /2 .

2. Potential energy: the energy possessed by a body by virtue of its position is called potential energy. This is given by mgh.

3. Law of conservation of energy: is a particle is acted upon by a conservative force system; the sum of the kinetic energy and potential energy is constant.

Energy stored in a spring: the energy stored in the spring is potential energy and is given by k s2/2. Where, k = spring constant, s =change in the length of the spring.Impulse: if a constant force P acts for a time t on a body, then impulse of the force = P.tMomentum: product of mass and the velocity is the linear momentum. Thus, Linear momentum = mv.Angular momentum: it is given by Torque.time where torque = T = I α.Relative velocity: two bodies moving in the same direction will have their relative velocity equal to the difference of their velocities. Thus, relative velocity will be u - v.Two bodies moving in opposite directions: the relative velocity in this case will be equal to – u +v.Angular velocity (ω): it is defined as the angle in radians turned through by a particle in unit time. Thus, ω = d θ/dt. If N is the revolutions per minute of a body then ω = 2 π N /

60. Tangential or linear velocity of a particle undergoing curvilinear motion and having angular velocity ω is given by v = ωr. And θ = ωt angular acceleration = α = v 2/ r.

Thick cylinder

Economic Order Quantity:Q =√2 AS/I Procurement cost/year = AS/QInventory carrying cost/year = I Q/2Average quantity held in inventory = Q/2.No of orders placed per year = Annual usage / EOQ.

Gear drive:Minimum No of teeth on pinion (Zp) = 14 for 20 0 stub teeth.

=31 for 14.5 0 full depth.= 17 for 20 0 full depth.

Cs = Service factor:= 1 for steady load.= 1.25 for light shock.= 1.5 for medium shock.= 1.8 for heavy shock.Tooth form factor (y):y = .124 -.684 / z for 14.5 0, v >20 m/s.y = .154 - .912/z for 20 0 full depth.y = .175 - .95/z for 20 0 stub teeth.Tangential tooth load, Ft = σd Cv b π y m.Tp / rp =Tg / rg, where Tp = P.60. 1000.Cs / 2 π N p.

σ d=design stress = 50 -80 N/mm2 for cast iron.=140 – 210 N/mm2 for Cast steel.=210 N/mm2 for Alloy steel.Velocity factor (Cv):= 3 / 3+V for ordinary gear v < 8 m/s.= 4.5 / 4.5 +V for carefully cut gear, v = 8 to 13 m/s.= 6 / 6+V for precision gear v = 13 to 20 m/s.= 5.55 / (5.55 + V) for high velocity gear v> 20 m/s.b=face width, generally taken as 9.5 to 12.5 mm.

Wear and Lubrication factor (Cw):

= 1.15 for good lubrication.= 1.25 for normal lubrication.= 1.35 for poor lubrication.

Mathematical Formulae:Circumference of a circle = 2 r.Diameter of circle = 2 r.Area of circle = r2 = D2/4.

Sin2 θ +cos2 θ = 1.Sec2 θ =1 + tan2 θ.Cosec2 θ = 1 + cot2 θ.

Sin (-θ) = - sin θ.Cos (-θ) = cosθ.

Sin (90 - θ) = sin θ.Cos (90 - θ) = sin θ.

Sin (90+ θ) = Cos θ.Cos (90 + θ) = - sin θ.

Sin (180 - θ) = sin θCos (180 - θ) = -cosθ.

Sin (180 + θ) = - sin θCos (180 + θ) = - cosθ

Sin (A+B) = SinA CosB + CosA SinB.Cos (A+B) =CosA CosB - SinA SinB.Sin (A-B) = SinA CosB - CosA SinB.Cos (A-B) = CosA CosB +SinA SinB.

SinC +SinD = 2 Sin (C+D)/2 .Cos (C-D)/2.SinC – SinD = 2 Cos (C+D)/2. Sin(C-D)/2.CosC +CosD = 2 cos(C+D)/2 .Cos (C-D)/2.CosC – CosD = 2 sin(C+D)/2. Sin (D-C)/2.

2 sinA cosB = sin (A+B) +sin (A-B).

2 CosA SinB = Sin (A+B) - Sin (A-B).2CosA CosB = Cos (A+B) + Cos (A-B).2 SinA SinB = Cos (A-B) - Cos (A-B).

tan (A+B) = (tanA + tanB) / (1- tanAtanB).tan (A-B) = (tanA - tanB) / (1+tanAtanB).

Sin 2A = 2 SinA CosA.Cos2A = Cos2A – Sin2A = 1 – 2Sin2A = 2Cos2A – 1.

Sin2A = 2 tanA / (1+tan2A).Cos2A = (1 – tan2A) / (1 + tan2A).tan2A = 2 tanA / (1- tan2A).

Sin3A = 3 SinA – 4Sin3A.Cos3A = 4Cos3A – 3CosA.tan3A = (3tanA- tan3A) / (1 – 3tan2A)

Sum of certain series:1. 1+2+3+4+ ……. +n = n (n+2) / 2.2. 12+22 +32 +42 + 52+ ……..+n2 = n (n+1) (2n+1) / 6.3. 1 3 + 2 3 +3 3 +4 3 + ………..+ n 3= [n (n+1)/2]2.4. 14 +24 + 34+ 44+………. + n4= n (n+1) (2n+1)(3n2 +3n -1)/30.5. Σ (2r-1) = 1+3 +5 +……… + (2n-1) = n 2.6. Σ (2r-1)2 =n (4n2-1)/3.7. Σ (2r-1)3 = n2 (2 n2-1).

% & + < = > ± µ ¼ ½ ¾ Π Σ η α λ μ ν ξ π ρ σ τ υ Ώ ≤ ≥ √ ∞ ← ↑ → θ ↓ ∂ ∆ ≈ ≠ β γ δ ε ζ Ф ω ω ω ω π ξ ν μ λ θ ζ ε δ δ δ γ β α Ω Π Σ ω σ φ Б Γ ,0 0 0 0 0 0 ± Ω ∂ µ

εθ =1/E (-ν )