Unit 1 / SPPU FE Engineering Mechanics

Unit 1: Force Systems — SPPU FE Engineering Mechanics

Topics: Resultant of forces, Varignon's theorem, centroid, and moment of inertia.

2024 pattern: end-sem Q212 marks in the 2024 end-sem

Which exam is this in?

2019 pattern

In-sem only (30 marks)

2024 pattern

End-sem Q2: solve any 2 of 4 sub-questions, 12 marks

Unit-by-unit question frequency and what repeats

Unit I — Force Systems + Centroid/MOI

In 2024 pattern: Q2 (12 marks). In 2019 pattern: in-sem only.

Sub-topic	Frequency	Notes
Resultant of concurrent force system	Every paper	Always involves resolving with angles
Resultant of parallel/general force system	Every paper	Square plate or mat foundation with 4 forces is the most recycled figure
Centroid of composite lamina	Every paper	Shaded area with cutout, circle or triangle removed, is the standard shape
Moment of Inertia (T, L, I section)	Every in-sem	Parallel + perpendicular axis theorem always needed
Varignon's theorem / Moments	Every in-sem	Usually appears as a short theory statement or a direct moment-of-components question

Recurring problem families and high-frequency variants

Square mat foundation with 4 column loads → find resultant w.r.t. origin O — May/Jun 2024, Nov/Dec 2025, May/Jun 2025

Concurrent force system with one unknown force or angle to make the resultant zero or the system stay in equilibrium — April 2025, November 2025, March 2025

Shaded composite lamina with a circular or triangular cutout → find centroid by tabular method — April 2025 and across multiple recent 2019-pattern papers

T-section, L-section, or I-section composite section → centroid first, then MOI by parallel axis theorem — recurring in recent in-sem papers

Sub-topics and how often they appear

Sub-topic	Priority	Frequency
Resultant of concurrent / parallel / general force system	High	Every paper
Centroid of composite lamina	High	Every paper
Moment of inertia of composite sections	Medium	Every in-sem, occasional in end-sem
Varignon's theorem / moments	Medium	Every in-sem

High — do this first, appears in every paper.

Medium — do this if you have time.

If you're on the 2024 pattern and Q2 gives you 4 sub-questions to pick 2 from, you want to be comfortable with at least 3 of the sub-topics above so you have a genuine choice.

High priority

Resultant of Force Systems

Concurrent force system

R_x = ∑F_x = ∑(F cos θ)

R_y = ∑F_y = ∑(F sin θ)

R = √(R_x² + R_y²)

θ = tan^-1(R_y / R_x)

Parallel force system

R = ∑F

x̄ = ∑M_O / R = ∑(F_ix_i) / ∑F_i

Varignon theorem

R × d = ∑(F_id_i)

Standard problem setup

Concurrent forces

Three or four forces act at a point. Each force is given as a magnitude and angle from horizontal, or as x and y components. Find the magnitude of the resultant and the angle it makes with the horizontal.

Step 1 — Resolve each force into x and y components.

Step 2 — Sum all x components to get R_x. Sum all y components to get R_y.

Step 3 — R = √(R_x² + R_y²).

Step 4 — θ = tan^-1(R_y / R_x). State the quadrant.

Parallel forces / mat foundation

A square plate or mat foundation has 4 loads acting at known positions. Find the resultant and its position from a reference point O.

Step 1 — R = sum of all forces. Watch the sign convention.

Step 2 — Take moments of all forces about O.

Step 3 — x̄ = ∑M_O / R to get the position from O.

Common mistakes

Angle reference confusion. A force at 30 degrees to the vertical is not the same as 30 degrees to the horizontal. Draw the angle before resolving.

Sign errors on R_y. Missing one negative flips the quadrant of the resultant.

Forgetting to state the quadrant after finding θ.

Using the wrong moment arm in a parallel force system. The arm must be the perpendicular distance from O to the line of action.

Most recycled version: square mat foundation with 4 column loads, find the resultant and where it acts with respect to corner O. This showed up in May/Jun 2024, Nov/Dec 2025, and May/Jun 2025.

High priority

Centroid of Composite Lamina

Formulas that appear

x̄ = ∑(A_ix̄_i) / ∑A_i

ȳ = ∑(A_iȳ_i) / ∑A_i

For cutouts: subtract the removed area and its moment contribution.

x̄ = (A_totalx̄_total - A_cutoutx̄_cutout) / (A_total - A_cutout)

Standard centroids you must know by heart

Shape	x̄ from left	ȳ from bottom
Rectangle (b × h)	b/2	h/2
Triangle (base b, height h)	b/3 from base vertex	h/3 from base
Circle (radius r)	centre	centre
Semicircle (radius r)	centre	4r / 3π from diameter
Quarter circle (radius r)	4r / 3π from each straight edge	4r / 3π

Standard problem setup

An L-shaped, T-shaped, or shaded lamina is given with dimensions. A circular or triangular portion is sometimes cut out. Find the centroid from a specified reference point.

Step 1 — Break the shape into simple parts: rectangles, triangles, circles.

Step 2 — For cutouts, list them as negative areas.

Step 4 — x̄ = ∑(A_ix̄_i) / ∑A_i, and the same for ȳ.

Always use a table. It keeps the work organised and examiners can follow where the marks should go even if the final answer is slightly off.

Common mistakes

Wrong centroid for a semicircle. It is 4r / 3pi from the diameter, not r/2.

Using local coordinates for one shape and global coordinates for another in the same table.

Forgetting the negative sign for cutouts.

Skipping the sketch and then measuring x_i or y_i from the wrong reference point.

Most recycled family: shaded composite lamina with a circular or triangular cutout, then a centroid table measured from one reference origin.

Medium priority

Moment of Inertia of Composite Sections

Basic MOI about centroidal horizontal axis

Shape	I_xx
Rectangle (b × d)	bd³ / 12
Triangle (base b, height h)	bh³ / 36
Circle (radius r)	πr⁴ / 4
Semicircle (radius r)	0.11r⁴

Parallel axis theorem

I = I_G + Ad²

Perpendicular axis theorem

I_z = I_x + I_y

Standard problem setup

A T-section, I-section, L-section, or channel section is given with dimensions. Find the moment of inertia about the centroidal x-x axis and sometimes the y-y axis.

Step 1 — Find the centroid of the composite section first.

Step 2 — For each part, write its own centroidal MOI.

Step 3 — Apply I = I_G + Ad² for each part.

Step 4 — Add all contributions carefully.

Common mistakes

Using the wrong axis with the base formula before applying the parallel axis theorem.

Calculating d from the base instead of from the sub-shape centroid to the overall centroid.

Trying to compute MOI before the overall centroid is known.

Forgetting that the semicircle value 0.11 r^4 is the memorised exam shortcut.

Most recycled family: T-section, L-section, or I-section MOI using centroid first, then the parallel axis theorem on each piece.

Medium priority

Varignon's Theorem

Formula

M_O(R) = ∑M_O(F_i)

In practice, instead of finding the perpendicular distance from O to a diagonal force, resolve the force into components at a convenient point on its line of action and take moments of each component separately.

Standard problem setup

A force of known magnitude acts along a diagonal or at an angle. Find its moment about a given point O.

Step 1 — Resolve the force into F_x and F_y at a convenient point on its line of action.

Step 2 — Moment of F_x about O = F_x × perpendicular distance.

Step 3 — Moment of F_y about O = F_y × perpendicular distance.

Step 4 — Total moment = moment of F_x + moment of F_y with the correct sign convention.

Common mistake: taking moment arms from the wrong point. The components must be taken at a point on the line of action of the original force. You cannot resolve at an arbitrary point.

Most recycled use: a diagonal or angled force where you resolve into horizontal and vertical components and then take moments about one reference point O instead of chasing the perpendicular distance directly.

MCQ Sampler

3 free concept checks for Unit 1

These questions cover the highest-frequency ideas only. The full bank has 50 questions: 3 free with open hints, then 47 premium questions with premium-only hints.

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Question 1

Concurrent Equilibrium

For a concurrent coplanar force system to be in equilibrium, which condition must hold?

Question 2

Composite Centroid with Hole

When a circular hole is removed from a uniform lamina, how should that area be treated in the centroid table?

Question 3

Parallel Axis Theorem

Which expression is the correct parallel axis theorem for one component of a composite section?

47 more questions stay locked. That premium set is where the locked hint layer and the harder-but-still-repeatable concept checks live.

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