Engineering mechanics is the backbone of every mechanical design — before any machine moves, stationary forces must be balanced and moving bodies must obey Newton's laws, making this topic non-negotiable for RRB JE Mechanical candidates.

Statics: forces in balance

Definition: Force — a push or pull that tends to change the state of rest or motion of a body. Measured in newtons (N).

Definition: Equilibrium — a body is in equilibrium when the net force and net moment (torque) acting on it are both zero (ΣF = 0; ΣM = 0).

A resultant force is the single equivalent force that produces the same effect as all the original forces combined. You find it by the parallelogram law (for two forces) or the polygon law (for many forces — draw them head-to-tail; the closing side is the resultant, reversed).

Varignon's Theorem (Principle of Moments): The moment of a resultant force about any point equals the algebraic sum of the moments of the component forces about the same point. This lets you break an awkward force into convenient horizontal and vertical components and sum their moments instead.

Lami's Theorem: When three concurrent, coplanar forces keep a body in equilibrium, each force is proportional to the sine of the angle between the other two:

F₁/sin α = F₂/sin β = F₃/sin γ

where α, β, γ are the angles opposite the respective forces (i.e., the angles between the other two forces). This is the fastest tool for three-force equilibrium problems on exams.

Free Body Diagram (FBD): Isolate the body; replace every contact and external agent with its force arrow. The FBD is the single most important tool in statics — draw it before writing any equation.

Centre of gravity and centroid

Definition: Centre of gravity (CG) — the point where the total weight of the body effectively acts. For uniform gravity, CG coincides with the centroid (geometric centre).

Key centroids to remember:

Triangle: at 1/3 height from base (h/3 from base).
Semi-circle: 4r/3π from the diameter.
Rectangle/Square: at the intersection of diagonals.

Moment of inertia (I): A measure of a cross-section's resistance to bending. For a rectangle (width b, depth h): I = bh³/12 about the centroidal axis. For a circle (diameter d): I = πd⁴/64.

Truss analysis

A truss is a structure of two-force members connected by pin joints. All loads apply at joints; no moments at joints.

Method of joints: Resolve forces at each joint (ΣFx = 0, ΣFy = 0). Start at a joint with only two unknowns.
Method of sections: Cut through the truss (through no more than 3 unknown members), isolate one side, and apply equilibrium. Faster when only one member's force is needed.

Dynamics: bodies in motion

Kinematics deals with motion without asking why (no forces). The three SUVAT equations for uniform acceleration:

v = u + at
s = ut + ½at²
v² = u² + 2as

where u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.

Newton's three laws:

A body stays at rest (or uniform motion) unless a net force acts on it.
Net force = mass × acceleration → F = ma.
Every action has an equal and opposite reaction.

Work-Energy Theorem

The net work done on a body equals its change in kinetic energy:

W_net = ΔKE = ½mv² − ½mu²

This is enormously useful: when friction is present, set the work done against friction equal to the loss of KE.

Momentum and Impulse

Definition: Momentum (p) = mass × velocity = mv (kg·m/s).

Impulse = F × t = Δp (change in momentum). A large force applied briefly, or a small force over a long time, can produce the same change in momentum — the principle behind airbags (increasing collision time reduces peak force on the occupant).

Conservation of momentum: In the absence of external forces, total momentum before collision = total momentum after. Applies to every collision type.

Friction

Definition: Friction — a contact force that opposes relative motion (or tendency of motion) between surfaces.

Static friction (f_s): Acts when the body is stationary. Can vary from zero up to a maximum: f_s(max) = μ_s × N, where μ_s is the coefficient of static friction and N is the normal reaction.
Kinetic (sliding) friction (f_k): Acts once the body slides: f_k = μ_k × N. Always f_k < f_s(max) for the same surfaces.

Angle of friction (φ): The angle the total reaction (N + friction vector resultant) makes with the normal. tan φ = μ.

Angle of repose (θ): The maximum slope angle at which a block rests without sliding. tan θ = μ_s. Numerically, angle of repose = angle of friction.

Inclined plane problems

For a block of mass m on a plane inclined at angle θ (coefficient of friction μ):

Component along plane (gravity): mg sin θ (downward along slope)
Normal reaction: N = mg cos θ
Friction force: f = μ mg cos θ (opposing motion)
Condition to slide: mg sin θ > μ mg cos θ → tan θ > μ

Screw jack: A screw jack converts torque to large lifting force. Efficiency depends on lead angle and friction angle. The self-locking condition (jack stays put when torque removed): lead angle < friction angle.

Belt friction (Euler's equation): T₁/T₂ = e^(μθ), where T₁ and T₂ are tensions on tight and slack sides, θ is the angle of wrap in radians. Exam favourite for belt-drive problems.

Simple Machines

A simple machine multiplies force (mechanical advantage, MA = Load/Effort) or changes its direction.

:::compare Simple Machines

Machine	Principle	MA	Example
Lever (Class 1)	Moment balance	>1, =1, or <1	See-saw, scissors
Pulley (movable)	Rope tension	2 per movable pulley	Block-and-tackle crane
Wheel & axle	Torque = F×r	R/r	Steering wheel, windlass
Inclined plane	Reduces effort	1/sin θ	Ramp, wedge
Screw	Fine threads = large MA	2πR/Lead	Screw jack, bolt
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Velocity ratio (VR) = distance moved by effort / distance moved by load.
Efficiency = MA/VR × 100%.

Worked Example

Question: A 10 kg block on a horizontal surface (μ = 0.3) is pushed by a horizontal force of 40 N. Find the acceleration. (g = 10 m/s²)

Solution:

Step 1: Normal reaction N = mg = 10 × 10 = 100 N.

Step 2: Kinetic friction f_k = μ_k × N = 0.3 × 100 = 30 N.

Step 3: Net force = Applied force − Friction = 40 − 30 = 10 N.

Step 4: F = ma → a = F/m = 10/10 = 1 m/s².

Conclusion: The block accelerates at 1 m/s² in the direction of the applied force.

Question: Three coplanar concurrent forces of 10 N, 20 N, and F are in equilibrium. The angle between the 10 N and 20 N forces is 150°, and between 20 N and F is 120°. Find F using Lami's Theorem.

Solution:

Step 1: Angle opposite to F (between the other two forces, 10 N and 20 N) = 150°.

Step 2: 10/sin 120° = 20/sin 90° = F/sin 150°.

Step 3: 20/sin 90° = 20/1 = 20. F = 20 × sin 150° = 20 × 0.5 = 10 N.

Conclusion: The third force F = 10 N.

Real-world example

Indian railways uses truss bridges (e.g., Howrah Bridge uses a cantilever truss) to span wide rivers with minimum material. Method of sections is the exact tool designers use to find the most stressed members. When a heavily loaded goods train crosses, every member must carry its design load — the equilibrium equations keep it standing.

Common misconception

Many students think friction always opposes motion. More precisely, static friction opposes the tendency of relative motion — it can point in any direction needed to keep a body stationary, including uphill along a slope. Only kinetic friction always acts exactly opposite to the direction of sliding.

:::keypoints Key points

Equilibrium requires ΣF = 0 AND ΣM = 0; always draw a Free Body Diagram first.
Lami's Theorem applies only to three concurrent, coplanar forces in equilibrium.
Kinematic equations (SUVAT) require constant acceleration.
Work-Energy Theorem: W_net = ΔKE — shortcut for problems involving variable forces or inclined surfaces.
Angle of repose = angle of friction; both equal arctan(μ).
Belt tension ratio: T₁/T₂ = e^(μθ); moment of inertia of rectangle: I = bh³/12.
Self-locking screw: lead angle < friction angle.
Efficiency = MA/VR; no machine is 100% efficient due to friction.
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:::memory
"For Delight, Seek Knowledge, With Momentum In Every Problem" → Forces, Dynamics, SUVAT, Kinematics, Work-energy, Momentum, Impulse, Equilibrium, Power.
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:::recap

Statics: resultant, Varignon's theorem, Lami's theorem, FBD, CG/centroid, MOI, truss methods.
Dynamics: SUVAT kinematics, Newton's laws F = ma, work-energy theorem, momentum-impulse.
Friction: static vs kinetic, angle of repose = angle of friction, inclined plane logic.
Belt friction and screw jack are direct exam numericals — memorise their formulas.
Simple machines: MA × efficiency = VR; each machine type has a characteristic advantage.
Always check units and convert (mm → m, kN → N) before plugging into formulas.
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Engineering Mechanics (Mech)