Statics: Forces, Equilibrium & Applications

TAKEAWAYS:

• Forces in equilibrium must balance in both magnitude and direction (sum of forces = 0).
• Moments (torques) cause rotation; equilibrium requires sum of moments = 0 about any point.
• Types of forces include tension, friction, weight, and normal reaction—each behaves differently in calculations.
• Stability depends on the center of gravity and base width; wider bases resist toppling.
• Free-body diagrams are essential tools to visualize and solve statics problems.
• Real-world applications include bridges, buildings, and even the human body’s skeletal system.

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1. Introduction to Statics

Statics is the branch of mechanics that studies objects at rest (or moving at constant velocity). Unlike dynamics (which deals with acceleration), statics focuses on forces in balance.

Key Concepts:

• Force: A push or pull acting on an object (measured in newtons, N).
• Equilibrium: When the net force and net moment on an object are zero, it remains at rest or moves uniformly.
• Types of Forces:
  • Weight (W): Force due to gravity ($W=mg$, where $m$ = mass, $g=9.8{\text{m/s}}^2$).
  • Tension (T): Force in a string, rope, or cable.
  • Normal Force (N): Perpendicular reaction force from a surface.
  • Friction (f): Opposes motion (static or kinetic).
  • Applied Force (F): Any external push/pull.

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2. Conditions for Equilibrium

For an object to be in complete equilibrium, two conditions must be met:

1. Translational Equilibrium: Sum of all forces in x-direction (ΣFₓ) = 0 and y-direction (ΣFᵧ) = 0.
2. Rotational Equilibrium: Sum of all moments (torques) about any point = 0.

Visualizing Forces: Free-Body Diagrams (FBD)

A free-body diagram is a sketch showing all forces acting on an object. Steps to draw an FBD:

1. Draw the object as a simple shape (e.g., a block, beam, or pendulum).
2. Label all forces with arrows (pointing away for pushes, toward for pulls).
3. Include angles if forces are not horizontal/vertical.

Example 1: A book of mass 2 kg rests on a table. Draw its FBD and find the normal force.

• Given: $m=2\text{kg}$, $g=9.8{\text{m/s}}^2$.
• Weight: $W=mg=2\times 9.8=19.6\text{N}$ (downward).
• Normal Force: Since the book is at rest, $N=W=19.6\text{N}$ (upward).

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3. Moments and Rotational Equilibrium

A moment (torque) is the rotational effect of a force.

• Moment (M) = Force × Perpendicular distance from the pivot. $M=F\times d$ (measured in N·m).
• Clockwise moments are negative; anticlockwise are positive.
• For equilibrium: ΣMoments = 0.

Example 2: A 5 m ladder leans against a wall. A 100 N force is applied at the top. Find the moment about the base.

• Given: $F=100\text{N}$, $d=5\text{m}$ (perpendicular distance).
• Moment: $M=100\times 5=500\text{N⋅m}$ (anticlockwise, so +500 N·m).

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4. Types of Equilibrium

Stability depends on:

• Center of Gravity (CG): The average position of an object’s weight.
• Base Width: Wider bases are more stable.

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5. Solving Statics Problems (Step-by-Step)

Problem: A 3 m uniform rod weighs 20 N. It is hinged at one end and supported by a 50 N force at the other end. Find the reaction force at the hinge. Solution:

1. Draw FBD:
   • Weight ($W=20\text{N}$) acts at the center (1.5 m).
   • Support force ($F=50\text{N}$) at the free end (3 m).
   • Hinge reaction ($R$) at the pivot (0 m).

2. Take moments about the hinge (ΣM = 0):

   • Moment due to weight: $20\times 1.5=30\text{N⋅m}$ (clockwise, -30 N·m).
   • Moment due to support: $50\times 3=150\text{N⋅m}$ (anticlockwise, +150 N·m).
   • Net moment: $-30+150=120\text{N⋅m}$ (not zero!).
   • Correction: The rod is not in equilibrium unless $R$ balances it.

3. Find $R$ using vertical forces (ΣFᵧ = 0): $R+50-20=0\Rightarrow R=-30\text{N}$ (downward).

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6. Applications of Statics

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7. Common Mistakes to Avoid

1. Ignoring friction: Always check if surfaces are rough/smooth.
2. Wrong pivot point: Choose a pivot to simplify moment calculations.
3. Sign errors: Clockwise moments are negative; anticlockwise are positive.
4. Assuming uniform distribution: Some objects (like rods) have weight acting at their center.

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Exam Tip

✅ NEB Exam Focus:

• Free-body diagrams are mandatory—always draw them!
• Moment calculations are frequent—practice with different pivots.
• Stability questions often involve center of gravity and base width.
• Units matter: Always use N (newtons) for forces and N·m for moments.
• Real-world problems: Expect questions on bridges, ladders, or beams.

Sample NEB-Style Questions:

1. A 10 N force acts at the end of a 2 m rod. Find the moment about a point 0.5 m from the end.
2. A 5 kg block rests on a table. If a 20 N horizontal force is applied, find the frictional force (given $\mu =0.3$).
3. Explain why a narrow-based object is less stable than a wide-based one.
4. Draw the FBD of a hanging sign with weight W, supported by two cables at angles.

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Summary:

• Equilibrium = No net force + No net moment.
• Moments = Force × Perpendicular distance.
• Stability = Low CG + Wide base.
• Always draw FBDs before solving!

Practice: Solve at least 5 problems from your textbook on ladders, beams, and pulleys to master this unit!