Engineering Mechanics for SSC JE Civil: Forces, Friction, Centroid, Moment of Inertia, Kinematics and Formulas with Solved Examples

Engineering mechanics is the foundation subject for all branches of engineering, and SSC JE (Civil and Mechanical) asks numerical and theory questions from it every year. This guide on engineering mechanics for SSC JE covers force systems, equilibrium, friction, the centroid and moment of inertia tables, kinematics, projectile motion and work and energy, with solved examples, practice questions and FAQs.

Force Systems

Friction

• Limiting friction F = μ N, where μ is the coefficient of friction and N is the normal reaction.
• Angle of friction: tan φ = μ; the angle of repose equals the angle of friction.
• Cone of friction: a cone with the semi-angle equal to the angle of friction.
• For a body on an inclined plane just about to slide, tan θ = μ.
• Static friction > kinetic friction; rolling friction < sliding friction.
• For a ladder in equilibrium against a smooth wall, the horizontal reaction at the wall equals the friction at the floor.

Centroid of Plane Figures

Moment of Inertia

• Parallel axis theorem: I = I_G + A h².
• Perpendicular axis theorem: I_z = I_x + I_y (for plane laminas).
• Radius of gyration: k = √(I/A); for a circle k = d/4.
• Section modulus Z = I / y_max.

Kinematics

Work, Power, Energy and Momentum

• Work = Force × displacement in the direction of force; the unit is the joule.
• Power = work / time; 1 horsepower = 746 watts.
• Kinetic energy = ½mv²; potential energy = mgh.
• Impulse = force × time = change in momentum.
• Law of conservation of momentum: in the absence of external forces, the total momentum remains constant.
• Coefficient of restitution e = (relative velocity of separation) / (relative velocity of approach); e = 1 for a perfectly elastic collision and 0 for a perfectly plastic collision.

Simple Machines

• Mechanical advantage (MA) = load / effort; velocity ratio (VR) = distance moved by effort / distance moved by load.
• Efficiency η = MA / VR.
• For an ideal machine MA = VR and η = 100%.
• A machine is self-locking if its efficiency is less than 50%.

Trusses

A frame is perfect if the number of members m = 2j − 3, where j is the number of joints. If m < 2j − 3 it is deficient, and if m > 2j − 3 it is redundant. The forces in the members are found by the method of joints or the method of sections.

Solved Examples

Example 1. Two forces of 10 N each act at right angles. Find the resultant.
R = √(100 + 100) = 14.14 N.

Example 2. If μ = 0.5, find the angle of friction.
φ = tan⁻¹(0.5) = 26.57°.

Example 3. A projectile is fired at 20 m/s at 30° to the horizontal. Find the range, maximum height and time of flight (g = 9.8).
R = 400 × sin 60° / 9.8 = 35.3 m; H = 400 × 0.25 / 19.6 = 5.1 m; T = 2 × 20 × 0.5 / 9.8 = 2.04 s.

Example 4. Find the moment of inertia of a rectangle of width 200 mm and depth 300 mm about its centroidal x-axis.
I = bd³/12 = 200 × 300³ / 12 = 4.5 × 10⁸ mm⁴.

Example 5. Find the number of members required for a perfect truss with 7 joints.
m = 2j − 3 = 14 − 3 = 11.

Quick Tips

• Memorise the centroid and MOI tables.
• Use the parallel axis theorem for composite sections.
• Remember that the maximum range of a projectile is at 45°.
• Check the perfect frame condition m = 2j − 3.

Quick Revision Points

• Lami’s theorem is for three concurrent forces in equilibrium.
• tan φ = μ.
• MOI of a rectangle about the centroid = bd³/12.
• I = I_G + Ah².
• A perfect frame has m = 2j − 3.

Practice Questions With Answers

1. What is the centroid of a triangle from its base?
   Answer: h/3
2. What is the MOI of a circle about its diameter?
   Answer: πd⁴/64
3. What is the angle of repose equal to?
   Answer: The angle of friction
4. At what angle is the range of a projectile maximum?
   Answer: 45°
5. How many members does a perfect truss with 5 joints have?
   Answer: 7
6. What is the radius of gyration of a circle?
   Answer: d/4
7. What is the unit of impulse?
   Answer: N s (kg m/s)
8. What is the MOI of a rectangle about its base?
   Answer: bd³/3
9. What is the efficiency of a machine?
   Answer: MA / VR
10. What does Lami’s theorem apply to?
    Answer: Three concurrent coplanar forces in equilibrium

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Civil Engineering notes page and the next topic Reinforced Concrete (RCC) Design.

Frequently Asked Questions

What is Lami’s theorem?

For three concurrent forces in equilibrium, each force is proportional to the sine of the angle between the other two.

What is the angle of friction?

The angle whose tangent equals the coefficient of friction.

What is the centroid?

The point at which the whole area of a plane figure can be assumed to be concentrated.

What is moment of inertia?

A measure of the resistance of a section to bending or rotation, I = Σ A y².

What is the parallel axis theorem?

I = I_G + Ah².

What is a perfect frame?

A truss with m = 2j − 3 members.

What is the coefficient of restitution?

The ratio of the relative velocity of separation to that of approach.

What is the range of a projectile?

R = u² sin 2θ / g.

What is a couple?

Two equal and opposite parallel forces that produce rotation.

Is engineering mechanics asked in SSC JE?

Yes. It is asked in both the civil and mechanical papers.

What is the SI unit of the moment of a force?

Newton metre.

What is self-locking in a machine?

A machine that holds the load without the effort; its efficiency is less than 50%.