Equilibrium of Forces | Jamb(UTME)

Jamb(utme) key points on equilibrium of coplanar forces; triangles and polygon of Forces; lami's theorem

Equilibrium of Coplanar Forces

1. Equilibrium occurs when the net force and net moment acting on a body are zero.
2. A body in equilibrium does not accelerate; it either remains at rest or moves with constant velocity.
3. Coplanar forces are forces acting in the same plane.
4. For a system of coplanar forces in equilibrium, the vector sum of all forces must be zero:
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   $\sum F_x = 0 \quad {and} \quad \sum F_y = 0$
5. The conditions for equilibrium of coplanar forces are:
   • The forces must form a closed polygon when represented as vectors.
   • The algebraic sum of horizontal forces $(F_x)$ must be zero.
   • The algebraic sum of vertical forces $(F_y)$ must be zero.
6. If three forces act on a body, they must be concurrent (intersect at a single point) for equilibrium.
7. The resultant of any two forces must be equal and opposite to the third force for equilibrium.
8. Coplanar forces can be resolved into components along perpendicular axes to simplify analysis.
9. Free-body diagrams (FBDs) are crucial for solving equilibrium problems.
10. Static equilibrium occurs when a body is at rest under the influence of balanced forces.
11. Dynamic equilibrium occurs when a body moves at constant velocity with no net force acting on it.
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Triangle of Forces

12. The triangle of forces method is used when three forces act on a body in equilibrium.
13. If three forces are in equilibrium, they can be represented as the sides of a triangle when drawn to scale.
14. The forces must form a closed triangle, meaning their vector sum is zero.
15. The direction of the forces must follow a consistent order (e.g., clockwise or counterclockwise) around the triangle.
16. The magnitude of the forces is proportional to the lengths of the corresponding sides of the triangle.
17. The triangle of forces is useful in solving problems involving inclined planes or tension in strings.
18. To construct a triangle of forces:
    • Draw one force to scale in its direction.
    • From the head of the first vector, draw the second force to scale.
    • Connect the tail of the first vector to the head of the second vector with the third force.
19. The triangle of forces is applicable only if the forces are coplanar and concurrent.
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Polygon of Forces

20. The polygon of forces extends the triangle method to systems with more than three forces.
21. If multiple forces act on a body in equilibrium, their vector sum must form a closed polygon.
22. The order of the vectors in the polygon must follow the direction of the forces.
23. To construct a polygon of forces:
    • Draw each force vector to scale and direction, starting from the endpoint of the previous force.
    • If the last vector closes the polygon by connecting back to the starting point, the forces are in equilibrium.
24. The polygon of forces helps analyze systems with more than three forces acting on a single point.
25. The method is often used in engineering and physics to study structures like bridges or cranes.
26. The closed polygon condition ensures that the resultant force is zero.
27. The polygon of forces also helps visualize the balance of forces in complex systems.
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Lami’s Theorem

28. Lami’s theorem applies to a body in equilibrium under the action of exactly three concurrent, coplanar forces.
29. The theorem states: $\frac{F_1}{\sin(\theta_1)} = \frac{F_2}{\sin(\theta_2)} = \frac{F_3}{\sin(\theta_3)}$ where:
    • $F_1, F_2, F_3$ are the magnitudes of the three forces.
    • $\theta_1, \theta_2, \theta_3$ are the angles opposite each force.
30. Lami’s theorem is derived from the triangle of forces and the sine rule in trigonometry.
31. It is applicable only when:
    • Exactly three forces act on the body.
    • The forces are concurrent and coplanar.
    • The body is in equilibrium.
32. Lami’s theorem simplifies solving equilibrium problems involving three forces.
33. To use Lami’s theorem:
    • Identify the three forces acting on the body.
    • Measure or calculate the angles between the forces.
    • Apply the formula to find unknown forces or angles.
34. Lami’s theorem is widely used in engineering, mechanics, and statics problems.
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Applications of Equilibrium Principles

In Everyday Life

35. A ladder resting against a wall is an example of equilibrium involving normal forces, friction, and weight.
36. A suspended traffic light is in equilibrium under the tension in the cables and its weight.
37. A book resting on a table is in static equilibrium, balanced by its weight and the table’s normal reaction.

In Engineering

38. Bridges are designed to ensure equilibrium under forces like tension, compression, and weight.
39. Cranes operate under equilibrium principles to lift and balance heavy loads.
40. Truss structures are analyzed using equilibrium equations to ensure stability.
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In Physics

41. Free-body diagrams are used to analyze forces acting on objects in physics problems.
42. Tension in strings is calculated using equilibrium principles in problems involving pulleys.
43. Inclined plane problems often involve resolving forces into components for equilibrium analysis.
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Key Observations

44. The equilibrium of forces ensures structures and objects remain stable and functional.
45. The triangle and polygon of forces provide geometric methods to analyze equilibrium.
46. Lami’s theorem is particularly useful for solving problems quickly when only three forces act.
47. Equilibrium principles are foundational in mechanics, statics, and engineering design.
48. The conditions of equilibrium can also be extended to rotational motion, ensuring no net torque acts on a body.
49. Understanding these principles helps design safe structures, predict motion, and solve mechanical problems effectively.
50. Mastering these concepts is crucial for fields like civil engineering, physics, and mechanics.
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Jamb(utme) key points on the conditions for equilibrium of rigid bodies under the action of parallel and nonparallel forces; centre of gravity and stability

Conditions for Equilibrium of Rigid Bodies

1. A rigid body is in equilibrium if it is at rest or moving with constant velocity.
2. There are two types of equilibrium: static equilibrium (body at rest) and dynamic equilibrium (body in motion).
3. For a rigid body to be in static equilibrium, the resultant force and resultant moment acting on it must be zero.
4. Translational equilibrium occurs when the sum of all forces acting on the body is zero.
5. Rotational equilibrium occurs when the sum of all moments about any point is zero.
6. The two conditions for equilibrium are:
   • $\Sigma F = 0$ (no net force)
   • $\Sigma M = 0$ (no net moment or torque)
7. Parallel forces act in the same or opposite directions along parallel lines of action.
8. For a body under parallel forces, equilibrium is achieved when:
   • The algebraic sum of all forces is zero.
   • The sum of moments about any point is zero.
9. Nonparallel forces act along different directions.
10. For nonparallel forces, equilibrium requires resolving forces into components and ensuring:
    • $\Sigma F_x = 0$ (no net force in the x-direction)
    • $\Sigma F_y = 0$ (no net force in the y-direction)
    • $\Sigma M = 0$ (no net torque about any point)
11. The line of action of forces is critical in determining the moments they create.
12. The moment of a force depends on the magnitude of the force and its perpendicular distance from the pivot point.
13. When multiple forces act, their vector sum must be zero for equilibrium.
14. The principle of moments states that clockwise moments must equal counterclockwise moments for rotational equilibrium.
15. A free-body diagram (FBD) helps visualize all forces acting on a body.
16. Reaction forces at supports must counteract applied forces for equilibrium.
17. Friction may be a factor in maintaining equilibrium, especially in inclined planes.
18. Equilibrium can be disturbed if the net force or net moment becomes nonzero.
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Centre of Gravity and Stability

19. The centre of gravity (CG) is the point where the weight of a body acts.
20. The CG is the average location of the weight distribution of a body.
21. For uniform objects, the CG is at the geometric center.
22. For irregular objects, the CG may lie outside the material of the body.
23. The position of the CG affects a body's stability.
24. A low CG increases stability, while a high CG reduces it.
25. To locate the CG, suspend the body from multiple points and mark the vertical lines through those points.
26. The CG lies at the intersection of these vertical lines.
27. The line of action of the weight passes through the CG.
28. For equilibrium, the line of action of the weight must fall within the base of support.
29. The wider the base of support, the more stable the body.
30. The stability of a body depends on the position of its CG relative to its base of support.
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Stability of Bodies

31. Stable equilibrium occurs when a body returns to its original position after being slightly displaced.
32. A body in stable equilibrium has its CG at the lowest possible position.
33. Examples of stable equilibrium include a cone resting on its base and a pendulum at its lowest point.
34. Unstable equilibrium occurs when a body moves further away from its original position after being slightly displaced.
35. A body in unstable equilibrium has its CG at the highest possible position.
36. Examples of unstable equilibrium include a cone balanced on its tip and a pencil standing on its end.
37. Neutral equilibrium occurs when a body remains in its new position after being displaced.
38. In neutral equilibrium, the CG remains at the same height during displacement.
39. Examples of neutral equilibrium include a ball on a flat surface and a cylinder lying horizontally.
40. The type of equilibrium is determined by the CG's movement relative to the base of support.
41. Stability is directly proportional to the area of the base of support.
42. Stability decreases as the height of the CG increases.
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Factors Affecting Stability

43. A lower CG makes a body more stable.
44. A wider base increases stability.
45. Increasing the weight of the base can improve stability by lowering the CG.
46. Stability can be improved by leaning towards the direction of an impending force (e.g., a climber leaning towards a wall).
47. A narrow base or higher CG makes a body easier to topple.
48. The angle of tilt at which the CG moves outside the base of support determines the tipping point.
49. Structures like skyscrapers use broad foundations to lower their effective CG and increase stability.
50. Engineers design objects, such as cars and chairs, with low CGs to enhance stability and prevent tipping.
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