Static equilibrium

Static Equilibrium: Key Concepts and Calculations

Introduction

  • Static equilibrium is a physical condition where an object remains at rest, and its center of mass does not change position.

Translational Equilibrium

  • Definition: An object is said to be in translational equilibrium when the sum of the forces acting through its center of mass equals zero.

  • Mathematical Representation:

    • ext{Net Force} = 0

    • This implies that both the sum of the forces in the X-direction (SF_x) and the sum of the forces in the Y-direction (SF_y) must also equal zero.

Rotational Equilibrium

  • Condition for Rotational Equilibrium:

    • An object is in rotational equilibrium if the sum of the moments (torques) about a reference point is equal to zero:

    • Mathematical Expression:

    • ext{Sum of Clockwise Moments} = ext{Sum of Anticlockwise Moments}

  • This can be summarized as:

    • ext{ΣM = 0}

  • Principle of Moments: The torque caused by a force depends on the magnitude of the force, the distance from the pivot point (lever arm), and the angle of application of the force.

Conditions for Equilibrium

  • For an object to be in equilibrium, the following conditions must be met:

    1. No net force is acting in any direction.

    2. No turning effect (rotation) about any point, ensuring rotational equilibrium.

Conditions for Static Equilibrium

  • Static Equilibrium Conditions:

    • ext{Net Force} = 0

    • ext{Net Torque/Moments} = 0

    • Specifics:

    • SF_x = 0

    • SF_y = 0

    • S_t = 0

    • S_{t_{cw}} = S_{t_{acw}}

Worked Examples

Example 2.3.1: Seesaw

  • Scenario: Two children on a seesaw.

  • Given:

    • Boy sits 1.50 m from pivot, mass = 20.0 kg

    • Girl's mass = 30.0 kg.

  • Assumption: Mass of the plank is negligible, and g = 9.80 \, ext{N/kg}.

    1. Calculate Force at Pivot:

    • Let FN be the force applied to the plank by the pivot point.

    1. Balance Calculation for Girl: Determine where the girl should sit to balance the boy.

Example 2.3.2: Seesaw at Reference Point

  • Task: Verify rotational equilibrium of seesaw with girl at point X.

  • Forces: Boy (196 N), Girl (294 N), Pivot force (490 N).

  • Analysis: Check ext{Sum of torques about X} = 0.

Example 2.3.3: Painter on Plank

  • Details: 70.0 kg painter stands 4.00 m from one end of a 6.00 m plank (plank mass = 20.0 kg).

  • Task: Find tension in left-hand rope (F_{t1}).

Example 2.3.4: Static Equilibrium of Cantilever

  • Setup: 18.0 m beam (30.0 kg), 10.0 m extension beyond supports.

  • Objective: Determine the force needed by right-hand support to maintain static equilibrium.

Review Questions

  1. What is required for rotational equilibrium?

    • Options may include no net torque, net torque acts, etc.

  2. Calculate position for balance on seesaw with known forces and masses.

  3. Determine maximum tension in various scenarios including ladder and beam examples.

Key Terms

  • Axis of Rotation: A pivot point around which moments occur.

  • Static Equilibrium: State of an object at rest with all forces and torques balanced.

  • Torque: A measure of the force causing an object to rotate about an axis.

  • Centre of Gravity: Point at which the gravitational force can be considered to act.

Conclusion

  • Static equilibrium requires a balance of forces and moments, a crucial concept in mechanics and engineering applications. This section has reviewed principles crucial for understanding and applying the concepts of equilibrium in practical situations.