Engineering Mechanics is a dependable, formula-light sub-topic of “Basics of Mechatronics” (Section A.2) in the GATE Robotics and Automation (RA) 2027 syllabus. It rewards clear free-body diagrams and equilibrium, a firm grip on friction (including belt-pulley), truss analysis, and the kinematics and dynamics of rigid bodies in plane motion. This guide reviews each area with its working relations and solves six GATE-style problems in full.
TABLE OF CONTENTS
Free-body diagrams & equilibrium
Every statics problem starts with a free-body diagram and the equilibrium equations: the sums of forces in two perpendicular directions and the sum of moments about any point all vanish. These three scalar equations solve for up to three unknown reactions in a planar system.
Friction & belt-pulley
Dry friction resists sliding up to a limit F = µN. A block on an incline is on the verge of sliding when the incline angle equals the angle of friction, tanφ = µ. For a belt or rope wrapping a pulley through angle θ, the tension ratio is the capstan (Euler) equation T1/T2 = eµθ.
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Trusses & frames
A truss carries load through two-force members in pure tension or compression. The method of joints applies force equilibrium at each pin; the method of sections cuts through members to expose the wanted force directly. A statically determinate plane truss satisfies m + r = 2j (members + reactions = twice the joints).
Kinematics & dynamics of rigid bodies
Plane kinematics links displacement, velocity and acceleration through v = u + at and s = ut + ½at², and for projectiles the range is R = u²sin2θ/g. Dynamics adds the cause of motion: Newton’s second law F = ma for translation and its rotational analogue T = Iα for rotation about a fixed axis.
Worked examples (GATE-style)
Example 1 — angle of friction
A block rests on an incline whose angle is slowly increased. If the coefficient of friction is 0.3, at what angle does the block start to slide?
Solution. Sliding begins when tanφ = µ = 0.3, so φ = tan−1(0.3).
φ = 16.7° (the angle of friction).
Example 2 — friction force on a flat surface
A 100 N block rests on a horizontal surface with µ = 0.3. Find the maximum friction force before it slides.
Solution. F = µN = 0.3 × 100 = 30 N (N equals the weight on a flat surface).
Example 3 — belt-pulley tension ratio
A belt wraps a pulley over a 180° (π rad) contact angle with µ = 0.25. Find the ratio of tight-side to slack-side tension.
Solution. T1/T2 = eµθ = e0.25×π = e0.785.
= 2.19.
Example 4 — projectile range
A projectile is launched at 20 m/s at 45°. Find its horizontal range (g = 9.81 m/s²).
Solution. R = u²sin2θ/g = 20² × sin90° / 9.81 = 400/9.81.
R = 40.77 m (maximum range occurs at 45°).
Example 5 — linear kinematics
A body starts from rest and accelerates at 2 m/s² for 5 s. Find its final velocity and the distance travelled.
Solution. v = u + at = 0 + 2×5 = 10 m/s.
s = ut + ½at² = 0 + ½×2×5² = 25 m.
Example 6 — rotational dynamics
A flywheel of moment of inertia 2 kg·m² is given an angular acceleration of 5 rad/s². Find the applied torque.
Solution. T = Iα = 2 × 5 = 10 N·m.
Key formulas
FORMULA SHEET
Equilibrium: ∑Fx = ∑Fy = ∑M = 0
Friction: F = µN, tanφ = µ
Belt: T1/T2 = eµθ
Kinematics: v = u + at, s = ut + ½at²
Projectile: R = u²sin2θ/g
Rotation: F = ma, T = Iα
Common mistakes to avoid
- Wrong normal force on an incline — N = mg·cosθ, not mg, so friction is µmg·cosθ.
- Using degrees inside eµθ — the wrap angle θ must be in radians.
- Forgetting the factor of 2 in sin2θ for projectile range.
- Mixing translational and rotational laws — use F = ma for linear, T = Iα for angular motion.
- Skipping the free-body diagram — most sign errors come from an incomplete FBD.
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Basics of Mechatronics — full solved-problem series
This guide is one part of the Section A.2 (Basics of Mechatronics) solved-problem series. Work through every sub-topic:
See also the umbrella guide, Basics of Mechatronics — Important Questions, and the complete GATE RA 2027 Syllabus.
Frequently asked questions
What is the angle of friction?
The angle of friction φ is the incline angle at which a body is just on the verge of sliding, given by tanφ = µ. For µ = 0.3 it is about 16.7°. Below this angle the block stays put; above it, it slides.
Why must the belt wrap angle be in radians?
The capstan equation T1/T2 = eµθ comes from integrating over the contact arc, where θ is measured in radians. Using degrees gives a wildly wrong tension ratio.
At what launch angle is projectile range maximum?
For a given speed on level ground, the range R = u²sin2θ/g is maximum at θ = 45°, because sin2θ peaks at 90°. The maximum range equals u²/g.
This solved-problems guide is part of the complete GATE RA 2027 Syllabus overview and the Basics of Mechatronics syllabus guide.
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