Important Formulas and Equations for the Mechanical Engineering Optional Subject

1. Strength of Materials:
- Stress = Force / Area
- Strain = Change in length / Original length
- Young's Modulus = Stress / Strain
- Shear Stress = Force / Area
- Shear Strain = Change in angle / Original angle
- Shear Modulus = Shear Stress / Shear Strain
- Bending Moment = Force x Distance
- Moment of Inertia = ∫(r^2) dm
- Deflection = (Force x Distance^3) / (3 x Young's Modulus x Moment of Inertia)

2. Thermodynamics:
- First Law of Thermodynamics: ΔU = Q - W
- Second Law of Thermodynamics: ΔS = Q/T
- Carnot Efficiency = 1 - (Tc/Th)
- Ideal Gas Law: PV = nRT
- Work Done by a Gas: W = PΔV
- Heat Transfer: Q = mcΔT

3. Fluid Mechanics:
- Bernoulli's Equation: P + (1/2)ρv^2 + ρgh = constant
- Reynolds Number: Re = (ρvd) / μ
- Darcy's Law: Q = -kA(ΔP/L)
- Pascal's Law: P = F/A
- Torricelli's Law: v = √(2gh)

4. Machine Design:
- Gear Ratio: N1/N2 = D2/D1 = ω1/ω2
- Efficiency: η = (Output Power / Input Power) x 100%
- Torsional Shear Stress: τ = (16T) / (πd^3)
- Bending Stress: σ = (Myc) / I
- Fatigue Strength: Sf = (Kf x Sp) / (Cf x Cg x Cs)

5. Manufacturing Processes:
- Machining Time: T = (L / f) + (L / (nN))
- Metal Removal Rate: MRR = (Volume of Material Removed) / (Time Taken)
- Cutting Speed: V = πDN
- Feed Rate: f = Nf
- Material Removal Rate: MRR = Vf

6. Industrial Engineering:
- Time Study Formula: Standard Time = (Basic Time) x (Performance Rating) x (Allowance Factor)
- Work Sampling Formula: Percentage of Time Spent on Task = (Number of Observations with Task) / (Total Number of Observations) x 100%
- Efficiency: Efficiency = (Actual Output / Standard Output) x 100%
- Productivity: Productivity = (Total Output / Total Input) x 100%

7. Engineering Mechanics:
- Newton's Second Law: F = ma
- Moment of a Force: M = Fd
- Friction Force: F = μN
- Center of Gravity: (Σm1r1) / (Σm1)
- Equilibrium: ΣF = 0, ΣM = 0

These are some of the important formulas and equations for the Mechanical Engineering optional subject. Remember to understand the concepts and derivations