Diploma - Engineering Mechanics - Equilibrium

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Diploma - Engineering Mechanics - Equilibrium. …

Diploma - Engineering Mechanics - Equilibrium.
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  • 1. Equilibrium Definition of 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 through it is acted upon by number of forces, it is said to be in equilibrium.” Analytical Condition of Equilibrium A For coplanar concurrent forces: we know that, resultant of concurrent forces is given by the R=     FX    2 formula, F   Y   2 Equilibrium means resultant force acting on the body is zero. i.e. R = 0    0  F  X  2  F   Y   2  Fx  0 and  Fy  0 Hence, analytical conditions of equilibrium for coplanar concurrent forces are i. ∑Fx = 0 i.e. Algebraic sum of component of all forces along x-axis must be equal to zero. ii. ∑Fy = 0 i.e. Algebraic sum of component of all forces along y-axis must be equal to zero. In other words, these
  • 2. analytical conditions of equilibrium can be started as follows: “If any numbers of forces acting on the body are in equilibrium, the algebraic sum of their components in two directions at right angle to each other must be equal to zero.” B. For coplanar non-concurrent forces: we know that, resultant of non-concurrent  R     forces is given by the formula 2          FX   FY 2     For body to be in equilibrium, the resultant force acting on it must be equal to zero. i.e. R=0 2 2 ∴ 0= ∴ ∑Fx = 0 and ∑Fy = 0   F   X     F   Y   According to varignon’s theorem of moments, the algebraic sum of moment of all forces about any point is equal to moment of the resultant force about the same point. i.e. As ∑M = R  x R=0 ∴ ∑M = 0 Therefore, Analytical conditions of equilibrium for coplanar non-concurrent forces are, i. ∑Fx = 0 ii. ∑Fy = 0 iii. ∑M = 0
  • 3. Concept of Free Body Diagram (F.B.D) In static, for considering the equilibrium of the body under the any system of forces each body is separated from its surroundings. Such body is known as free body. If all active and reactive forces acting on a free body are shown, the diagram is known as free body diagram. Lami’s Theorem (Equilibrium Of Three Coplanar Concurrent Forces) Statement: Its states, “If three forces acting at a point on a body keep it at rest, then each force is proportional to the sine of the angle between the other two forces.”
  • 4. Mathematically, Notations: P = Q = R sinα sinβ sinγ α = Angle between Q and R β = Angle between P and R γ = Angle between P and Q P, Q and R = Three concurrent forces. Proof: To prove that, P = Q = R sinα sinβ sinγ Construct a parallelogram OACB such that OA = P, OB = Q, and OC = R. In ∆ OAC, applying sine rule, we get, OA = AC = OC sin  π - α sin  π - β sin  π - γ 
  • 5. ∴ P = Q = R = K where K is a constant. sinα sinβ sinγ ∴ P = Ksinα, Q = Ksinβ and R = Ksinγ ∴ P α sin , Q α sinβ and R α sinγ Join Ednexa for free online live lectures for Engineering/Diploma For more information visit www.ednexa.com Or call 9011041155.