Equilibrium Of Forces
Edwin Ogie Library
Equilibrium of Forces E‑Note
Objectives:
• Apply the conditions for equilibrium of coplanar forces.
• Use triangle and polygon laws of forces and Lami’s theorem to solve problems.
• Analyze moments of forces and couples (torque) with applications.
• Resolve forces into perpendicular directions and determine resultants and equilibrants.
• Differentiate between stable, unstable, and neutral equilibrium.
Page 1: Introduction to Equilibrium of Forces
Equilibrium in mechanics occurs when the sum of forces and moments acting on a body is zero. This e‑note explains the equilibrium of particles and rigid bodies, the principles of moments, and the concepts of centre of gravity and stability.
Page 2: Equilibrium of Particles – Coplanar Forces
A particle is in equilibrium when the vector sum of all coplanar forces acting on it is zero. Mathematically, if forces F1, F2, …, Fn act on a particle, then:
ΣF = 0
This requires both horizontal and vertical components to sum to zero.
Example CP1: Two Force Equilibrium
Solution:
For two forces to balance, they must be equal in magnitude and opposite in direction.
Example CP2: Three Force Equilibrium
Solution:
For three coplanar forces to be in equilibrium, the vector sum is zero. (Diagram and vector resolution can be applied.)
Example CP3: Resolving Forces into Components
Solution:
Resolve each force into horizontal and vertical components. Sum of horizontal components = 0 and sum of vertical components = 0.
Page 3: Equilibrium of Particles – Triangles and Polygon of Forces
When forces are arranged head-to-tail, they form a closed polygon if the particle is in equilibrium. A triangle of forces is a special case where three forces acting on a particle can be represented by the sides of a triangle.
Example TP1: Triangle of Forces
Solution:
If three forces of known magnitudes form a closed triangle when arranged head-to-tail, they are in equilibrium.
Example TP2: Force Polygon – Four Forces
Solution:
For four forces to be in equilibrium, their force polygon must be a closed quadrilateral.
Example TP3: Using Graphical Methods
Solution:
Draw the forces to scale using the head-to-tail method; if the final vector closes the polygon, equilibrium is achieved.
Page 4: Equilibrium of Particles – Lami’s Theorem
Lami’s Theorem applies to a particle in equilibrium under the action of three non-parallel coplanar forces. It states that:
F1/sin α = F2/sin β = F3/sin γ
where α, β, and γ are the angles between the forces.
Example LT1: Applying Lami’s Theorem
Solution:
Given forces F1 and F2 and the angles between them, determine F3 using the theorem.
Example LT2: Finding an Unknown Force
Solution:
With F1 = 30 N, F2 = 40 N, and appropriate angles, calculate F3 using F1/sin α = F3/sin γ.
Example LT3: Verifying Equilibrium with Lami’s Theorem
Solution:
Check if the given forces satisfy Lami’s theorem; if they do, the particle is in equilibrium.
Page 5: Principles of Moments – Moment of a Force
The moment (or torque) of a force is the product of the force and the perpendicular distance from the pivot to the line of action of the force. It is given by:
Moment = Force × Perpendicular Distance
Example M1: Basic Moment Calculation
Solution:
A force of 50 N acts at a perpendicular distance of 0.8 m from the pivot; Moment = 50 × 0.8 = 40 N·m.
Example M2: Moment with Angled Force
Solution:
Resolve the force into a component perpendicular to the lever arm; if the effective perpendicular force is 30 N at 1 m, Moment = 30 N·m.
Example M3: Balancing Moments
Solution:
For equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.
Page 6: Principles of Moments – Moment of a Couple
A couple consists of two equal and opposite forces whose lines of action do not coincide. The moment of a couple (or torque) is given by:
Moment of Couple = Force × Distance between Forces
Example MC1: Calculating a Couple’s Moment
Solution:
Two forces of 20 N separated by 0.5 m produce a moment = 20 × 0.5 = 10 N·m.
Example MC2: Equilibrium of a Couple
Solution:
In a balanced couple, the clockwise and anticlockwise moments cancel.
Example MC3: Practical Application of a Couple
Solution:
For a wrench applying a force at a distance, calculate the torque to loosen a bolt.
Page 7: Equilibrium of Rigid Bodies – Resolution and Composition
To analyze forces on rigid bodies, forces are resolved into two perpendicular components (usually horizontal and vertical). The vector sum of these components gives the resultant force.
Resultant = √(Fx² + Fy²)
Example RC1: Resolving a Force
Solution:
A force of 100 N at 30° above horizontal: Fx = 100 cos 30° ≈ 86.6 N, Fy = 100 sin 30° = 50 N.
Example RC2: Composing Two Perpendicular Forces
Solution:
Forces: 40 N horizontally and 30 N vertically. Resultant = √(40² + 30²) = 50 N.
Example RC3: Finding the Direction of the Resultant
Solution:
Angle = tan−1(Fy/Fx) = tan−1(30/40) ≈ 36.87° above horizontal.
Page 8: Equilibrium of Rigid Bodies – Resultant and Equilibrant
The resultant of several forces is the single force which has the same effect as the original forces when applied at the same point. The equilibrant is a force equal in magnitude but opposite in direction to the resultant.
Example RE1: Finding the Resultant
Solution:
For forces 30 N east and 40 N north, Resultant = √(30² + 40²) = 50 N; direction = tan−1(40/30) ≈ 53.13° north of east.
Example RE2: Finding the Equilibrant
Solution:
Equilibrant is equal in magnitude to the resultant but opposite in direction. Thus, if the resultant is 50 N at 53.13° north of east, the equilibrant is 50 N at 53.13° south of west.
Example RE3: Combined Force System
Solution:
Resolve a system of three forces into their components and compute the resultant; then determine the equilibrant.
Page 9: Centre of Gravity and Stability
The centre of gravity of a body is the point at which the weight of the body is considered to be concentrated. A body is said to be in stable equilibrium if, when displaced, it returns to its original position; unstable if it moves further away; and neutral if it remains in the new position.
Example CG1: Finding the Centre of Gravity of a Uniform Rod
Solution:
For a uniform rod, the centre of gravity is at its midpoint.
Example CG2: Stability of a Suspended Object
Solution:
An object suspended from its centre of gravity is in neutral equilibrium; if suspended away, it will be unstable.
Example CG3: Analysis of a Tipped Object
Solution:
Examine the shifting of the centre of gravity relative to the base of support to determine stability.
Page 10: Worked Examples – Equilibrium of Forces (Summary)
This page recaps the earlier sections with additional numerical examples combining multiple concepts.
Example E1: Combined Equilibrium Problem
Solution:
Using forces in equilibrium and the principle of moments, solve for an unknown force.
Example E2: System of Forces in Equilibrium
Solution:
Resolve the forces acting on a particle and verify that the vector sum is zero.
Example E3: Lami’s Theorem in a Real-World Scenario
Solution:
Apply Lami’s theorem to a system of three forces to determine an unknown force.
Page 11: Summary of Key Formulas and Concepts
- Equilibrium of Forces: ΣF = 0; components in both x and y directions must vanish.
- Cosine Rule: Used for triangles of forces.
- Lami’s Theorem: F1/sin α = F2/sin β = F3/sin γ
- Moment of a Force: M = F × d
- Moment of a Couple: M = F × distance
- Resultant Force: √(ΣFx² + ΣFy²)
- Equilibrant: Equal in magnitude but opposite to the resultant
- Centre of Gravity: The point where the weight is effectively concentrated
- Stable, Unstable, Neutral Equilibrium: Defined by the position of the centre of gravity relative to the base
Page 12: Extended Discussion and Applications
Equilibrium of forces is fundamental in engineering and physics. It is used to design structures, analyze mechanical systems, and ensure stability in various applications. The methods outlined here help in determining unknown forces, designing stable structures, and verifying that structures can withstand applied loads.
Page 13: Additional Tips and Insights
Practice by drawing vector diagrams for multiple forces and checking equilibrium conditions. Use Lami’s theorem as a quick check for three-force systems, and always ensure moments balance about any chosen pivot.
Page 14: Final Summary
- Equilibrium of particles requires the vector sum of forces to be zero.
- Forces can be represented graphically as triangles or polygons.
- Lami’s theorem provides a relation between three concurrent forces in equilibrium.
- Moments (torques) are crucial for the equilibrium of rigid bodies.
- Resultant and equilibrant forces are determined by resolving forces into perpendicular components.
- Centre of gravity and stability dictate whether an object will return to its original position when displaced.
Page 15: Quiz Introduction and Final Review
Review all the key concepts: conditions for equilibrium, moments, resultant forces, and stability. Test your understanding with the quiz below.
30 CBT JAMB Quiz on Equilibrium of Forces
Click the "Start Quiz" button to begin. You will have 15 minutes to answer 30 questions.
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