Engineering Mechanics — Dynamics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

GATE Weightage: ~5–8 marks/year (Mechanical branch); often combined with statics in a single 2-mark question.

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Kinematics of Particles

• Rectilinear motion: x(t), v = dx/dt, a = dv/dt = d²x/dt²
• Projectile motion: Range R = (u² sin 2θ)/g, Max height H = (u² sin²θ)/(2g), Time of flight T = (2u sinθ)/g
• Curvilinear motion: Separate into horizontal and vertical components

Newton’s Laws & Force Analysis

• F = ma — core equation; identify all forces, draw Free Body Diagram (FBD)
• Weight = mg downward; normal force perpendicular to surface; friction f ≤ μN
• Common mistake: Forgetting that weight changes with g in different planet problems

Work-Energy & Impulse-Momentum

• Work-Energy: W = ΔKE = ½m(v² – u²)
• Potential energy: gravitational PE = mgh; spring PE = ½kx²
• Conservation of energy: Total mechanical energy constant if no non-conservative forces
• Impulse-Momentum: J = Δp = ∫F dt; for constant force: J = F·Δt = m(v – u)
• Conservation of momentum: External force = 0 → m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Collisions

Type	e (coefficient of restitution)	KE after collision
Perfectly elastic	e = 1	KE conserved
Inelastic	0 < e < 1	KE lost
Perfectly inelastic	e = 0	Bodies stick together

• e = (v₂ – v₁)/(u₁ – u₂) = relative speed of separation / relative speed of approach

Rigid Body Rotation

• Angular displacement θ (rad), ω = dθ/dt, α = dω/dt
• Torque: τ = Iα (analogous to F = ma)
• Moment of inertia I = Σmr² for discrete; I = ∫r² dm for continuous
• Rotational KE = ½Iω²
• Angular momentum L = Iω; τ = dL/dt

Key Formulas Summary

Linear:    v = u + at,    s = ut + ½at²,    v² = u² + 2as
Projectile: R = u²sin2θ/g,   H = u²sin²θ/2g
Collision: v₁' = (m₁–em₂)/(m₁+m₂)u₁ + (1+e)m₂/(m₁+m₂)u₂
Rotation:  τ = Iα,   KE = ½Iω²,   L = Iω

⚡ GATE Tips

• Always draw FBD before writing equations — 50% of mistakes happen here
• In collision problems, check if momentum AND energy are simultaneously conserved (elastic = both, inelastic = only momentum)
• For rigid body rotation about a fixed axis, parallel axis theorem: I = I_cm + md²

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Kinematics of Particles

Kinematics describes motion without considering its causes. Rectilinear motion uses:

• Position: x(t)
• Velocity: v = dx/dt
• Acceleration: a = dv/dt = d²x/dt²

Integration gives: v = u + at, s = ut + ½at², v² = u² + 2as

Projectile motion decomposes into:

• Horizontal: aₓ = 0 → x = u cosθ · t
• Vertical: aᵧ = –g → y = u sinθ · t – ½gt²

Common mistake: Students forget that maximum range (R = u²/g) occurs at θ = 45° only for level ground. For elevated target, angle differs.

Newton’s Laws and Force Analysis

First Law (Inertia): Body continues at rest/uniform motion unless acted upon by external force.

Second Law: F = ma (vector equation — resolve components)

Third Law: Action–reaction pairs act on different bodies.

Free Body Diagram (FBD) — Critical for GATE

1. Isolate the body
2. Show ALL external forces with arrows
3. Never show internal forces or reaction forces on other bodies

Work-Energy Theorem

Work done by all forces = Change in kinetic energy:

W_total = ½mv² – ½mu²

For conservative forces, work is path-independent and W = –ΔPE.

Gravitational spring potential energy:

• PE_gravity = mgh (height measured from chosen datum)
• PE_spring = ½kx² (x = displacement from natural length)

Conservation of mechanical energy: E_total = KE + PE = constant (no friction/drag)

Impulse-Momentum Theorem

Impulse J = ∫F dt = F_avg · Δt = Δp (change in momentum)

For impulse of constant force: J = F·Δt

Key insight: Average force during collision = Δp/Δt. Higher Δt → lower average force (why airbags work).

Conservation of Momentum

Valid when net external force = 0:

Σ p_initial = Σ p_final

This is fundamental in collision analysis and rocket propulsion.

Collisions in One Dimension

Coefficient of restitution: e = relative speed of separation / relative speed of approach

• e = 1: Perfectly elastic (KE conserved)
• e = 0: Perfectly inelastic (bodies stick, maximum KE loss)
• 0 < e < 1: Partially inelastic (most real collisions)

Velocities after collision:

v₁ = [(m₁ – em₂)u₁ + (1+e)m₂u₂] / (m₁+m₂)
v₂ = [(m₂ – em₁)u₂ + (1+e)m₁u₁] / (m₁+m₂)

Lost KE in perfectly inelastic collision: ΔKE = ½μ(v₁ – v₂)²(1 – e²), where μ = m₁m₂/(m₁+m₂)

Rigid Body Rotation

Angular kinematic equations (analogous to linear):

• ω = ω₀ + αt
• θ = ω₀t + ½αt²
• ω² = ω₀² + 2αθ

Moment of inertia depends on axis of rotation:

• I about any axis = I_cm + md² (parallel axis theorem)
• Thin rod (about center): I = mL²/12; about end: I = mL²/3
• Solid cylinder/disc: I = mR²/2
• Solid sphere: I = (2/5)mR²

Torque and angular momentum:

• τ = Iα
• L = Iω (angular momentum)
• τ = dL/dt ( rotational analogue of F = dp/dt)

Work-Energy for rotation:

• W = τθ
• KE_rot = ½Iω²
• Power P = τω

Example Problem

A block of mass 2 kg slides down a 30° incline (μ = 0.3) from height 5 m. Find its speed at the bottom.

Solution:

• Height h = 5 m, angle θ = 30°, μ = 0.3
• Forces along incline: mg sinθ – f = ma
• Normal force N = mg cosθ
• Friction f = μN = μmg cosθ
• So: mg sinθ – μmg cosθ = ma
• a = g(sinθ – μ cosθ) = 9.81(0.5 – 0.3×0.866) = 9.81(0.5 – 0.2598) = 9.81×0.2402 = 2.356 m/s²
• Using v² = u² + 2as, s = h/sinθ = 5/0.5 = 10 m
• v² = 0 + 2×2.356×10 = 47.12 → v = 6.86 m/s

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Advanced Kinematics — Curvilinear Motion

For motion in a plane with curvilinear path, use normal and tangential components:

• Tangential acceleration: aₜ = d|v|/dt (changes speed)
• Normal acceleration: aₙ = v²/ρ (changes direction; ρ = radius of curvature)
• Total acceleration: a = √(aₜ² + aₙ²)

For circular motion (ρ = R = constant):

• aₙ = v²/R = ω²R
• In uniform circular motion: aₜ = 0, a = aₙ only

Variable Mass Systems — Rocket Propulsion

Rocket equation (Tsiolkovsky):

• Thrust = v_e · (dm/dt) where v_e = exhaust velocity relative to rocket
• Net external force on rocket: F_ext = M(dv/dt) + v_e(dM/dt)
• For rocket in gravity-free space, F_ext = 0: M(dv/dt) = –v_e(dM/dt)

GATE Tip: Variable mass problems require careful identification of system boundary. The expelled/added mass is NOT part of the main system when applying F = ma.

Impulse-Momentum in 2D and 3D

Impulse-momentum theorem is vector-valued:

• J = Δp = p_final – p_initial
• Components: Jₓ = Δpₓ, Jᵧ = Δpᵧ, J_z = Δp_z
• In 2D collisions, both x and y momentum components conserve independently

Oblique Collisions

For off-center (oblique) collisions:

1. Resolve velocities into normal and tangential components
2. Along tangent: velocity components are unchanged (no impulsive force in tangent direction)
3. Along normal: apply 1D collision formula using e
4. Reconstruct final velocity vectors from new normal + unchanged tangential components

System of Rigid Bodies

Energy methods vs. momentum methods:

Method	Conservative Forces	Non-conservative/Impulsive
Work-Energy	ΔKE = W_net	ΔKE = W_non-conservative
Momentum	Σp conserved only if F_ext = 0	Apply impulse equations

Angular impulse-momentum: ∫τ dt = ΔL (angular impulse = change in angular momentum)

Radius of Gyration

Defined by k such that I = mk². Useful when tables give k instead of I.

Rolling Motion (No Slip)

Condition: v_cm = ωR (translational and rotational linked)

Kinetic energy of rolling:

• KE_total = ½m v_cm² + ½I_cm ω²
• Substituting ω = v_cm/R: KE = ½mv_cm²(1 + I/(mR²))
• For solid cylinder (I = mR²/2): KE = ¾mv_cm²
• For hollow cylinder (I = mR²): KE = mv_cm²

Acceleration down incline (no slip):

• a = (g sinθ) / (1 + I/(mR²))
• Solid sphere: a = (5/7)g sinθ
• Hollow sphere: a = (3/5)g sinθ
• Solid cylinder: a = (2/3)g sinθ

GATE Common Mistake: Students often confuse the moment of inertia formula for different objects. Always derive I for the given axis, or use the parallel axis theorem correctly.

Damping and Forced Oscillations (Rotational)

For torsional vibrations:

• Equation: I·d²θ/dt² + c·dθ/dt + k·θ = T(t)
• Damping ratio: ζ = c/(2√(Ik))
• Critical damping: c_cr = 2√(Ik); ζ = 1
• Natural frequency: ω_n = √(k/I)

GATE Exam Strategy — Dynamics

Question types to expect:

1. Kinematics graph problems — area under a-t, v-t graphs
2. Block-on-incline — friction, energy, acceleration
3. Collision — find e or final velocities (1D)
4. Rotational KE and moment of inertia — composite bodies
5. Projectile from height — maximum range, time of flight

Common GATE mistakes to avoid:

• Using g = 10 m/s² instead of 9.81 when precision matters
• Mixing up work-energy and conservation of momentum conditions
• Incorrectly applying parallel axis theorem
• Forgetting sign conventions for PE in energy conservation

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