Equilibrium of forces

General physics

🎯 Learning Objectives

By the end of this lesson, you will be able to:
• Understand the concept of equilibrium for coplanar forces acting on a point
• Draw and interpret vector triangles to represent forces in equilibrium
• Apply the triangle rule for vector addition to solve equilibrium problems
• Calculate unknown forces using trigonometry and vector triangle methods
• Analyze real-world situations involving forces in equilibrium using vector triangles
• Solve complex equilibrium problems by constructing accurate vector diagrams

🗣️ Language Objectives

Students will develop their physics communication skills by:
• Using precise scientific terminology when describing force equilibrium and vector relationships
• Explaining vector triangle construction and interpretation using appropriate technical vocabulary
• Reading and interpreting equilibrium problems written in English with confidence
• Communicating mathematical solutions and vector analysis clearly in written English
• Understanding and using directional language (resultant, components, coplanar) accurately in vector contexts

📚 Key Terms and Translations

English Term	Russian Translation	Kazakh Translation
Equilibrium	Равновесие	Тепе-теңдік
Vector triangle	Треугольник векторов	Векторлық үшбұрыш
Coplanar forces	Компланарные силы	Бір жазықтықтағы күштер
Resultant force	Равнодействующая сила	Тең әсерлі күш
Vector addition	Сложение векторов	Векторларды қосу
Triangle rule	Правило треугольника	Үшбұрыш ережесі
Force components	Компоненты силы	Күш құраушылары
Concurrent forces	Пересекающиеся силы	Қиылысатын күштер

🃏 Vocabulary Study Cards

Vector Triangle

Definition: A geometric representation where three vectors form a closed triangle

Key Property: For equilibrium, vectors form a closed triangle with zero resultant

Application: Used to find unknown forces in equilibrium problems

Example: Three forces 5N, 7N, and 9N acting at angles can form a triangle

Equilibrium of Forces

Definition: State where all forces acting on an object have zero net effect

Condition: Vector sum of all forces equals zero

Characteristic: Object remains at rest or moves with constant velocity

Example: A book resting on a table experiences equilibrium

Coplanar Forces

Definition: Forces that all lie in the same plane

Property: Can be represented using 2D vector diagrams

Analysis: Simplified using x and y components

Example: Forces acting on a object sliding down an inclined plane

Triangle Rule

Definition: Method for adding vectors by placing them head-to-tail

Process: Draw vectors consecutively to form a triangle

Result: Third side represents the resultant vector

Application: Essential for solving force equilibrium problems

📖 Glossary of Terms

Equilibrium

A state in which all forces acting upon an object are balanced, resulting in no net force and no acceleration. The object either remains at rest or continues to move at constant velocity.

Translation

Russian: Равновесие — это состояние, при котором все силы, действующие на объект, сбалансированы, что приводит к отсутствию результирующей силы и ускорения. Объект либо остается в покое, либо продолжает двигаться с постоянной скоростью.

Kazakh: Тепе-теңдік — бұл затқа әсер ететін барлық күштердің теңгерілген күйі, бұл тең әсерлі күштің және үдеудің болмауына әкеледі. Зат не тыныштықта қалады, не тұрақты жылдамдықпен қозғалуын жалғастырады.

Vector Triangle

A graphical method of representing three vectors where they are arranged to form a closed triangle. In equilibrium problems, if three forces form a closed triangle when placed head-to-tail, they are in equilibrium with zero resultant.

Translation

Russian: Треугольник векторов — это графический метод представления трех векторов, где они расположены так, чтобы образовать замкнутый треугольник. В задачах равновесия, если три силы образуют замкнутый треугольник при соединении конец к началу, они находятся в равновесии с нулевой результирующей.

Kazakh: Векторлық үшбұрыш — үш векторды көрсетудің графикалық әдісі, мұнда олар жабық үшбұрыш құру үшін орналастырылады. Тепе-теңдік есептерінде үш күш басы мен аяғын қосқанда жабық үшбұрыш құрса, олар нөлдік тең әсерлі күшпен тепе-теңдікте болады.

Coplanar Forces

Forces that all lie within the same geometric plane. These forces can be completely described using two-dimensional coordinate systems and their effects can be analyzed using 2D vector methods.

Translation

Russian: Компланарные силы — это силы, которые все лежат в одной геометрической плоскости. Эти силы могут быть полностью описаны с использованием двумерных систем координат, и их эффекты могут быть проанализированы с помощью методов двумерных векторов.

Kazakh: Бір жазықтықтағы күштер — барлығы бір геометриялық жазықтықта жататын күштер. Бұл күштерді екі өлшемді координаталық жүйелер арқылы толық сипаттауға болады және олардың әсерлерін 2D векторлық әдістер арқылы талдауға болады.

Resultant Force

The single force that would produce the same effect as all the individual forces acting together. It is found by vector addition of all the component forces. In equilibrium, the resultant force is zero.

Translation

Russian: Равнодействующая сила — это единственная сила, которая производила бы тот же эффект, что и все отдельные силы, действующие вместе. Она находится путем векторного сложения всех составляющих сил. В равновесии равнодействующая сила равна нулю.

Kazakh: Тең әсерлі күш — барлық жеке күштердің бірге әсер етуімен бірдей әсер көрсететін жалғыз күш. Ол барлық құраушы күштердің векторлық қосындысы арқылы табылады. Тепе-теңдікте тең әсерлі күш нөлге тең.

Triangle Rule

A method for adding vectors geometrically by placing them head-to-tail in sequence. The resultant vector is drawn from the tail of the first vector to the head of the last vector, completing the triangle.

Translation

Russian: Правило треугольника — это метод геометрического сложения векторов путем размещения их последовательно конец к началу. Результирующий вектор проводится от начала первого вектора к концу последнего вектора, завершая треугольник.

Kazakh: Үшбұрыш ережесі — векторларды геометриялық қосудың әдісі, оларды дәйекті түрде басы мен аяғын жалғау арқылы. Тең әсерлі вектор бірінші вектордың аяғынан соңғы вектордың басына дейін сызылып, үшбұрышты аяқтайды.

🔬 Theory: Vector Triangles and Force Equilibrium

Understanding Force Equilibrium

When an object is in equilibrium, the vector sum of all forces acting on it equals zero. This fundamental principle means that there is no net force to cause acceleration, and the object either remains at rest or continues moving with constant velocity.

For coplanar forces acting at a point, equilibrium can be analyzed using vector methods. The most effective graphical method is the construction of vector triangles.

Translation

Russian: Когда объект находится в равновесии, векторная сумма всех сил, действующих на него, равна нулю. Этот фундаментальный принцип означает, что нет результирующей силы для создания ускорения, и объект либо остается в покое, либо продолжает двигаться с постоянной скоростью.

Kazakh: Зат тепе-теңдікте болғанда, оған әсер ететін барлық күштердің векторлық қосындысы нөлге тең. Бұл іргелі принцип үдеу туғызатын тең әсерлі күштің жоқтығын білдіреді және зат не тыныштықта қалады, не тұрақты жылдамдықпен қозғалуын жалғастырады.

The Vector Triangle Method

A vector triangle is formed when three forces in equilibrium are arranged head-to-tail. The key principle is that if three concurrent forces are in equilibrium, they can be represented by the three sides of a closed triangle.

To construct a vector triangle:

1. Choose a suitable scale for representing force magnitudes

2. Draw the first force vector with correct direction and magnitude

3. From the head of the first vector, draw the second force vector

4. From the head of the second vector, draw the third force vector

5. For equilibrium, the third vector should return to the starting point

Translation

Russian: Треугольник векторов образуется, когда три силы в равновесии расположены голова к хвосту. Ключевой принцип заключается в том, что если три пересекающиеся силы находятся в равновесии, они могут быть представлены тремя сторонами замкнутого треугольника.

Kazakh: Векторлық үшбұрыш тепе-теңдіктегі үш күш басы мен аяғын жалғап орналастырылғанда құрылады. Негізгі принцип — егер үш қиылысатын күш тепе-теңдікте болса, оларды жабық үшбұрыштың үш қабырғасымен көрсетуге болады.

Mathematical Analysis of Vector Triangles

Once a vector triangle is constructed, trigonometry can be used to find unknown force magnitudes or angles. The most commonly used relationships are:

Sine Rule:
a/sin(A) = b/sin(B) = c/sin(C)

Cosine Rule:
c² = a² + b² — 2ab cos(C)

Where a, b, and c are the side lengths (representing force magnitudes), and A, B, and C are the opposite angles in the triangle.

Translation

Russian: После построения треугольника векторов можно использовать тригонометрию для нахождения неизвестных величин сил или углов. Наиболее часто используемые соотношения включают правило синусов и правило косинусов.

Kazakh: Векторлық үшбұрыш құрылғаннан кейін белгісіз күш шамаларын немесе бұрыштарды табу үшін тригонометрияны пайдалануға болады. Ең жиі қолданылатын қатынастар синус ережесі мен косинус ережесін қамтиды.

Theory Questions

Easy Question: What is the condition for three forces to be in equilibrium when represented as a vector triangle?

Answer

For three forces to be in equilibrium when represented as a vector triangle, the triangle must be closed. This means that when the three force vectors are arranged head-to-tail, the third vector must return exactly to the starting point of the first vector, forming a complete triangle with no gaps.

Medium Question: Three forces of magnitudes 8N, 6N, and 10N act at a point. Can they be in equilibrium? Justify your answer using the triangle inequality.

Answer

Yes, these three forces can be in equilibrium. For three forces to form a triangle (and thus be in equilibrium), they must satisfy the triangle inequality: the sum of any two sides must be greater than the third side.

Checking all combinations:
— 8 + 6 = 14 > 10 ✓
— 8 + 10 = 18 > 6 ✓
— 6 + 10 = 16 > 8 ✓

Since all conditions are satisfied, these forces can form a triangle and therefore can be in equilibrium. In fact, these numbers form a right triangle (6² + 8² = 10²), so the forces would be in equilibrium when arranged at right angles.

Medium Question: A force of 15N acts horizontally to the right, and another force of 20N acts at 60° above the horizontal. What third force is needed for equilibrium?

Answer

To find the third force, we need to find the resultant of the first two forces, then the third force will be equal and opposite to this resultant.

First, resolve the 20N force:
— Horizontal component: 20 × cos(60°) = 20 × 0.5 = 10N (right)
— Vertical component: 20 × sin(60°) = 20 × 0.866 = 17.32N (up)

Total components of first two forces:
— Horizontal: 15 + 10 = 25N (right)
— Vertical: 0 + 17.32 = 17.32N (up)

Resultant magnitude: √(25² + 17.32²) = √(625 + 300) = √925 = 30.4N
Resultant direction: tan⁻¹(17.32/25) = 34.7° above horizontal

Therefore, the third force needed is 30.4N at 34.7° below the horizontal (or 214.7° from the positive x-axis).

Hard Question (Critical Thinking): A bridge design engineer claims that for a suspension bridge, the tension forces in three cable segments meeting at a joint will always form a vector triangle for equilibrium. However, a colleague argues that this is only true under specific geometric constraints. Analyze both viewpoints considering: (a) the physical constraints of cable systems, (b) the mathematical requirements for equilibrium, and (c) the practical implications for bridge design safety.

Answer

Analysis of Both Viewpoints:

(a) Physical Constraints of Cable Systems:
The engineer is correct that forces must form a vector triangle for equilibrium, but the colleague correctly identifies geometric constraints:

— Cables can only support tension (not compression), limiting force directions
— Cable angles are constrained by tower positions and bridge geometry
— Cables cannot have infinite flexibility — minimum bending radii exist
— Material properties limit maximum tensions

(b) Mathematical Requirements:
The mathematical requirement (∑F = 0) means forces MUST form a closed vector triangle for equilibrium. However:

— Not all combinations of three forces can form a valid triangle (triangle inequality)
— Cable directions are not arbitrary — they’re fixed by geometry
— The force magnitudes must adjust to satisfy both equilibrium AND geometric constraints

(c) Practical Design Implications:
Both engineers are partially correct:

Supporting the first engineer:
— Equilibrium absolutely requires vector triangle closure
— This is a fundamental design principle that cannot be violated

Supporting the colleague:
— Geometric constraints may make equilibrium impossible with certain cable arrangements
— Poor geometry can require unrealistically high tensions
— Safety factors must account for dynamic loads, not just static equilibrium

Critical Conclusion:
The first engineer is mathematically correct but oversimplified. The colleague correctly identifies that successful bridge design requires the geometric configuration to allow reasonable force magnitudes while maintaining equilibrium. Both static equilibrium (vector triangle) AND geometric feasibility must be satisfied simultaneously.

Design Implication:
Engineers must design cable geometry first to ensure equilibrium is possible with acceptable forces, rather than assuming any three cables will automatically work.

💪 Exercises on Memorization of Terms

Complete the Definitions

1. When three forces are in equilibrium, they can be represented by the sides of a _______ _______.

Answer

Vector triangle

2. Forces that all lie in the same plane are called _______ forces.

Answer

Coplanar forces

3. The _______ _______ is the single force that would produce the same effect as all individual forces combined.

Answer

Resultant force

4. For equilibrium, the vector sum of all forces must equal _______.

Answer

Zero

5. The method of adding vectors by placing them head-to-tail is called the _______ _______.

Answer

Triangle rule

6. When forces meet at a single point, they are called _______ forces.

Answer

Concurrent forces

🎥 Educational Video Resource

Additional Video Resources:

• Physics Online: Vector Addition and Equilibrium

• Khan Academy: Forces and Newton’s Laws

• Professor Dave: Vector Addition Graphically

📐 Worked Problem Examples

Example 1: Simple Vector Triangle

Problem: Three forces act at a point: 12N horizontally to the right, 16N at 60° above the horizontal, and a third unknown force. If the system is in equilibrium, find the magnitude and direction of the third force.

Vector triangle diagram showing three forces in equilibrium

Step-by-Step Solution

Given:
— Force 1: F₁ = 12N horizontally to the right
— Force 2: F₂ = 16N at 60° above horizontal
— Force 3: F₃ = unknown (for equilibrium)

Step 1: Resolve F₂ into components
— F₂ₓ = 16 × cos(60°) = 16 × 0.5 = 8N (right)
— F₂ᵧ = 16 × sin(60°) = 16 × 0.866 = 13.86N (up)

Step 2: Find total components of F₁ and F₂
— Total x-component: 12 + 8 = 20N (right)
— Total y-component: 0 + 13.86 = 13.86N (up)

Step 3: For equilibrium, F₃ must balance these components
— F₃ₓ = -20N (left)
— F₃ᵧ = -13.86N (down)

Step 4: Calculate magnitude and direction of F₃
— |F₃| = √(20² + 13.86²) = √(400 + 192.1) = √592.1 = 24.3N
— θ = tan⁻¹(13.86/20) = tan⁻¹(0.693) = 34.7°

Answer: The third force is 24.3N at 34.7° below the horizontal to the left (or 214.7° from the positive x-axis).

Verification: The three forces form a closed triangle, confirming equilibrium.

Example 2: Using Sine Rule in Vector Triangle

Problem: Three forces in equilibrium form a triangle where two forces are 15N and 20N with an angle of 120° between them. Find the third force using the vector triangle method.

Complete Solution Using Trigonometry

Given:
— Force A = 15N
— Force B = 20N
— Angle between A and B = 120°

Step 1: Use cosine rule to find the third force (Force C)
The angle between the forces is 120°, so in the vector triangle, the angle opposite to force C is 120°.

C² = A² + B² — 2AB cos(120°)
C² = 15² + 20² — 2(15)(20)cos(120°)
C² = 225 + 400 — 600(-0.5)
C² = 625 + 300 = 925
C = √925 = 30.4N

Step 2: Find angles in the triangle using sine rule
sin(A)/a = sin(B)/b = sin(C)/c

Where A, B, C are angles opposite to sides a, b, c respectively.

sin(α)/15 = sin(120°)/30.4
sin(α) = (15 × sin(120°))/30.4 = (15 × 0.866)/30.4 = 0.427
α = sin⁻¹(0.427) = 25.2°

sin(β)/20 = sin(120°)/30.4
sin(β) = (20 × 0.866)/30.4 = 0.570
β = sin⁻¹(0.570) = 34.8°

Verification: α + β + 120° = 25.2° + 34.8° + 120° = 180° ✓

Answer: The third force is 30.4N, and the angles in the vector triangle are 25.2°, 34.8°, and 120°.

Example 3: Complex Equilibrium with Multiple Forces

Problem: A ring is suspended by three strings making angles of 30°, 45°, and 60° with the vertical. If the tensions in the first two strings are 50N and 70N respectively, find the tension in the third string and the weight of the ring.

Advanced Vector Analysis Solution

Given:
— T₁ = 50N at 30° from vertical
— T₂ = 70N at 45° from vertical
— T₃ = unknown at 60° from vertical
— Weight W = unknown (vertically downward)

Step 1: Set up coordinate system (x-axis horizontal, y-axis vertical up)

Step 2: Resolve all forces into components
T₁ components:
— T₁ₓ = 50 sin(30°) = 50 × 0.5 = 25N (assume right)
— T₁ᵧ = 50 cos(30°) = 50 × 0.866 = 43.3N (up)

T₂ components:
— T₂ₓ = -70 sin(45°) = -70 × 0.707 = -49.5N (left)
— T₂ᵧ = 70 cos(45°) = 70 × 0.707 = 49.5N (up)

T₃ components:
— T₃ₓ = T₃ sin(60°) = T₃ × 0.866 (assume left, so negative)
— T₃ᵧ = T₃ cos(60°) = T₃ × 0.5 (up)

Weight:
— Wₓ = 0
— Wᵧ = -W (down)

Step 3: Apply equilibrium conditions
Horizontal equilibrium (ΣFₓ = 0):
25 — 49.5 — T₃ × 0.866 = 0
-24.5 = T₃ × 0.866
T₃ = 24.5/0.866 = 28.3N

Step 4: Find weight using vertical equilibrium (ΣFᵧ = 0)
43.3 + 49.5 + 28.3 × 0.5 — W = 0
43.3 + 49.5 + 14.15 = W
W = 106.95N ≈ 107N

Verification using vector triangle:
The three tension forces should form a closed triangle with the weight completing the equilibrium.

Answer:
— Tension in third string: T₃ = 28.3N
— Weight of ring: W = 107N

Note: The direction assumption for T₃ was corrected by the negative result indicating the opposite direction.

🧪 Interactive Investigation - PhET Simulation

Explore vector addition and equilibrium using this interactive simulation:

Investigation Tasks:

Task 1: Create three vectors that form a closed triangle. Record their magnitudes and directions. Verify that they represent forces in equilibrium.

Task 2: Given two vectors, use the simulation to find the third vector needed to create equilibrium. Compare your graphical result with calculations.

Task 3: Investigate how changing the magnitude of one vector affects the required equilibrium vector. What patterns do you observe?

Task 4: Try to create equilibrium with vectors that violate the triangle inequality. What happens?

Investigation Answers and Analysis

Task 1 Analysis:
Students should find that any three vectors forming a closed triangle represent equilibrium forces. The key insight is that the resultant (sum) of the three vectors equals zero when they form a closed shape.

Task 2 Verification:
The graphical method should match calculations within measurement accuracy. Students learn that both approaches yield the same result, reinforcing the connection between mathematical and graphical vector analysis.

Task 3 Pattern Recognition:
Students observe that:
— Increasing one vector requires adjustment of the equilibrium vector
— The triangle must always close for equilibrium
— Certain combinations become impossible (triangle inequality violations)

Task 4 Understanding Limitations:
When triangle inequality is violated (sum of two sides ≤ third side), equilibrium is impossible. This demonstrates physical constraints in real force systems — not all force combinations can achieve equilibrium.

Key Learning Outcome:
Students develop intuition for vector equilibrium and understand both the mathematical requirements and physical limitations of force systems.

👥 Collaborative Group Activity

Work with your team to complete this interactive vector equilibrium challenge:

Group Challenge: Design a Force System

Vector Triangle Engineering Challenge

Challenge: Your team must design a three-cable support system for a hanging sculpture that weighs 500N. The cables must meet specific architectural constraints.

Constraints:

Cable 1: Must make 30° with the vertical
Cable 2: Must make 45° with the vertical
Cable 3: Angle is flexible but must be between 20° and 70° from vertical
No cable tension can exceed 300N
System must be in perfect equilibrium

Deliverables:

Complete vector triangle diagram with all forces labeled
Mathematical analysis showing equilibrium
Verification that all constraints are satisfied
Alternative design if constraints cannot be met
Team presentation (5 minutes maximum)

Alternative Group Activities:

• Vector Triangle Race: Teams compete to solve equilibrium problems fastest using both graphical and analytical methods

• Real-World Analysis: Analyze force systems in bridges, playground equipment, or architectural structures

• Design Optimization: Minimize maximum cable tension for various hanging load configurations

📝 Individual Assessment - Structured Questions

Question 1: Vector Triangle Analysis

Three forces act at a point in equilibrium: 25N at 0°, 30N at 120°, and a third force F₃. (a) Construct the vector triangle and find F₃ analytically. (b) If the 30N force direction changes to 130°, recalculate F₃ and analyze how the 10° change affects the equilibrium system.

Answer

Part (a): Original configuration

Method 1 — Component analysis:
Force 1: F₁ = 25N at 0° (horizontal right)
— F₁ₓ = 25N, F₁ᵧ = 0N

Force 2: F₂ = 30N at 120°
— F₂ₓ = 30cos(120°) = 30(-0.5) = -15N
— F₂ᵧ = 30sin(120°) = 30(0.866) = 26N

For equilibrium: F₃ₓ = -(25-15) = -10N, F₃ᵧ = -26N
|F₃| = √(10² + 26²) = √(100 + 676) = √776 = 27.9N
Direction: θ = tan⁻¹(26/10) = 68.96° below negative x-axis = 248.96°

Method 2 — Cosine rule verification:
Using cosine rule in the triangle: F₃² = 25² + 30² — 2(25)(30)cos(60°)
F₃² = 625 + 900 — 1500(0.5) = 775
F₃ = 27.8N ✓

Part (b): Modified configuration (30N at 130°)
Force 2: F₂ = 30N at 130°
— F₂ₓ = 30cos(130°) = 30(-0.643) = -19.3N
— F₂ᵧ = 30sin(130°) = 30(0.766) = 23N

For equilibrium: F₃ₓ = -(25-19.3) = -5.7N, F₃ᵧ = -23N
|F₃| = √(5.7² + 23²) = √(32.5 + 529) = √561.5 = 23.7N
Direction: θ = tan⁻¹(23/5.7) = 76.1° below negative x-axis

Analysis:
The 10° change reduced F₃ from 27.9N to 23.7N (15% decrease) and significantly changed its direction. This demonstrates the sensitivity of equilibrium systems to small angular changes.

Question 2: Equilibrium Constraint Analysis

A mass m is suspended by three strings attached to fixed points forming an equilateral triangle. Each string makes the same angle θ with the vertical. (a) Derive a general expression for the tension T in each string in terms of m, g, and θ. (b) Calculate the minimum angle θ if each string can support a maximum tension of 2mg. (c) Analyze what happens to the system as θ approaches 90°.

Answer

Part (a): General expression for tension

Due to symmetry, all three strings have equal tension T and make equal angles θ with vertical.

Vertical equilibrium:
3T cos(θ) = mg
Therefore: T = mg/(3cos(θ))

Horizontal equilibrium:
The horizontal components of the three tensions must sum to zero, which is automatically satisfied due to the 120° symmetry.

Part (b): Minimum angle calculation

Given constraint: T ≤ 2mg
mg/(3cos(θ)) ≤ 2mg
1/(3cos(θ)) ≤ 2
1 ≤ 6cos(θ)
cos(θ) ≥ 1/6
θ ≤ cos⁻¹(1/6) = 80.4°

Therefore, θ_min = 80.4°

Part (c): Analysis as θ → 90°

As θ approaches 90°:
— cos(θ) → 0
— T = mg/(3cos(θ)) → ∞

Physical interpretation:
As the strings become nearly horizontal, the vertical component of tension approaches zero, requiring infinite tension to support the weight. This is physically impossible, demonstrating that there’s a maximum angle beyond which equilibrium cannot be maintained with finite forces.

Critical insight:
This analysis reveals a fundamental limitation in cable/string systems — very small vertical components require impractically large tensions, which is why bridge cables and similar structures are designed with adequate vertical angles.

Question 3: Vector Triangle Construction and Optimization

Design a vector triangle for three forces where: (a) one force is fixed at 40N horizontally, (b) the second force has a magnitude between 20N and 60N, and (c) the angle between the first two forces can be varied. Determine the range of possible magnitudes for the third force and identify the configuration that minimizes the third force. Provide both analytical and graphical analysis.

Answer

Given constraints:
— F₁ = 40N (horizontal, fixed)
— F₂ = 20N to 60N (variable magnitude)
— Angle α between F₁ and F₂ = variable
— Find range of F₃ and minimize F₃

Analytical approach using cosine rule:
F₃² = F₁² + F₂² — 2F₁F₂cos(α)
F₃ = √(1600 + F₂² — 80F₂cos(α))

Part (a): Range analysis

Case 1: F₂ = 20N
— Maximum F₃ (α = 180°): F₃ = √(1600 + 400 + 1600) = √3600 = 60N
— Minimum F₃ (α = 0°): F₃ = √(1600 + 400 — 1600) = √400 = 20N

Case 2: F₂ = 60N
— Maximum F₃ (α = 180°): F₃ = √(1600 + 3600 + 4800) = √10000 = 100N
— Minimum F₃ (α = 0°): F₃ = √(1600 + 3600 — 4800) = √400 = 20N

Overall range: 20N ≤ F₃ ≤ 100N

Part (b): Optimization for minimum F₃

To minimize F₃, we differentiate with respect to α:
dF₃/dα = (80F₂sin(α))/(2√(1600 + F₂² — 80F₂cos(α)))

Setting dF₃/dα = 0: sin(α) = 0, so α = 0° or 180°

Checking second derivative: minimum occurs at α = 0°

At α = 0°:
F₃ = |40 — F₂|

For F₂ between 20N and 60N:
— When F₂ = 40N: F₃ = 0 (impossible physically)
— Minimum practical F₃ = 20N when F₂ = 20N or 60N

Graphical verification:
Vector triangle analysis confirms that:
— Parallel forces (α = 0°) give minimum resultant
— Antiparallel forces (α = 180°) give maximum resultant
— Minimum occurs when F₁ and F₂ are collinear

Optimal configuration:
F₂ = 20N at α = 0° (same direction as F₁) gives F₃ = 20N
Alternative: F₂ = 60N at α = 0° also gives F₃ = 20N

Physical interpretation:
The minimum third force occurs when the first two forces are aligned, requiring only enough force to balance their resultant.

Question 4: Advanced Equilibrium System Analysis

A triangular framework consists of three rigid bars connected by pin joints, with external forces applied at each joint. The system must remain in equilibrium under the applied loads. Given that forces of 100N (downward), 150N (30° above horizontal), and 120N (45° below horizontal) are applied at the three vertices, analyze: (a) whether equilibrium is possible, (b) the internal forces in each bar, (c) the stability of the system if one bar's stiffness is reduced by 50%. Consider both static equilibrium and structural stability.

Answer

Part (a): Equilibrium possibility analysis

Given forces:
— F₁ = 100N downward (270°)
— F₂ = 150N at 30° above horizontal
— F₃ = 120N at 45° below horizontal (315°)

Check vector sum for equilibrium:

F₁ components: F₁ₓ = 0, F₁ᵧ = -100N

F₂ components: F₂ₓ = 150cos(30°) = 150(0.866) = 129.9N
F₂ᵧ = 150sin(30°) = 150(0.5) = 75N

F₃ components: F₃ₓ = 120cos(315°) = 120cos(-45°) = 120(0.707) = 84.8N
F₃ᵧ = 120sin(315°) = 120sin(-45°) = 120(-0.707) = -84.8N

Total components:
ΣFₓ = 0 + 129.9 + 84.8 = 214.7N ≠ 0
ΣFᵧ = -100 + 75 — 84.8 = -109.8N ≠ 0

Conclusion: The given forces do NOT sum to zero, so equilibrium is impossible without additional constraint forces from supports.

Part (b): Internal forces analysis

Since the external forces don’t balance, the framework must be supported. Assuming pin supports that provide reaction forces to achieve equilibrium:

Required reaction forces:
— Horizontal reaction: -214.7N
— Vertical reaction: +109.8N
— Total reaction magnitude: √(214.7² + 109.8²) = √58157.4 = 241.2N

Internal bar forces (using method of sections):
For a statically determinate triangular frame, each bar carries tension/compression to maintain joint equilibrium. The internal forces depend on:
— Bar orientations (triangle geometry)
— Applied loads at joints
— Support reactions

Without specific triangle geometry, exact values cannot be calculated, but the method involves:
1. Calculate support reactions for overall equilibrium
2. Apply equilibrium at each joint
3. Solve for bar forces using joint equilibrium equations

Part (c): Stability analysis with reduced stiffness

Structural considerations:

Static stability: Remains unchanged if one bar’s stiffness reduces by 50%, as this affects deformation but not equilibrium force distribution (assuming linear elastic behavior).

Dynamic stability: Reduced stiffness affects:
— Natural frequencies (lower frequencies)
— Damping characteristics
— Response to dynamic loads
— Buckling resistance (if bars are in compression)

Critical factors:
— If the reduced-stiffness bar carries compression, buckling risk increases
— System becomes more flexible, increasing deflections
— Resonance frequencies shift, potentially causing dynamic instability

Engineering implications:
1. **Load redistribution:** Forces remain the same, but deformations increase
2. **Failure modes:** Buckling becomes more likely in the weakened member
3. **Dynamic response:** System becomes more susceptible to oscillations
4. **Safety factors:** Must be increased to account for reduced stiffness

Recommendation:
The 50% stiffness reduction significantly compromises structural integrity. Analysis should include:
— Buckling analysis for compression members
— Dynamic analysis for oscillatory loads
— Progressive collapse assessment
— Alternative load paths in case of member failure

Question 5: Synthesis and Critical Analysis

A mechanical engineer claims that "any three forces can be brought into equilibrium by adjusting their directions, regardless of their magnitudes." A physicist disagrees, arguing that "magnitude relationships impose fundamental constraints that cannot be overcome by directional adjustments alone." Evaluate both positions by: (a) providing mathematical proof or counterexample, (b) analyzing the geometric constraints in vector triangles, (c) discussing practical implications for engineering design, and (d) exploring the boundary conditions where equilibrium becomes impossible.

Answer

Position Evaluation: The physicist is correct.

Part (a): Mathematical proof through counterexample

Triangle Inequality Constraint:
For three forces F₁, F₂, F₃ to form a vector triangle (equilibrium condition), they must satisfy:
- F₁ + F₂ > F₃
- F₁ + F₃ > F₂
- F₂ + F₃ > F₁

Counterexample:
Consider forces: F₁ = 10N, F₂ = 15N, F₃ = 40N

Checking triangle inequality:
- 10 + 15 = 25 10N + 15N.

Mathematical proof:
In any triangle, the length of one side cannot exceed the sum of the other two sides. This is a geometric impossibility, not a directional issue.

Part (b): Geometric constraints analysis

Vector triangle requirements:
1. **Closure condition:** Vectors must form a closed polygon
2. **Triangle inequality:** No side can exceed sum of other two
3. **Existence condition:** For given magnitudes, specific angle relationships must be satisfied

Constraint visualization:
Given F₁ and F₂, the possible values of F₃ form a range:
|F₁ - F₂| ≤ F₃ ≤ F₁ + F₂

Outside this range, no directional adjustment can achieve equilibrium.

Part (c): Engineering design implications

Supporting the physicist's position:

Design constraints:
- Load magnitudes are often fixed by functional requirements
- Available force magnitudes may be limited by component capabilities
- Economic factors limit over-sizing of components

Practical examples:
1. **Bridge cables:** If main load exceeds sum of support capabilities, no cable arrangement can work
2. **Crane design:** Counterweight limitations impose fundamental stability constraints
3. **Structural joints:** Member strength limits cannot be overcome by orientation changes

Engineering methodology:**strong>
1. First check magnitude compatibility (triangle inequality)
2. Then optimize orient