Equilibrium of a Rigid Body, Moments and Center of Gravity

1. Moments and Torque

References: NCERT Physics Class 11, Chapter 7: System of Particles and Rotational Motion

• Moment of a force:
• Definition: The moment of a force is the turning effect produced by the force about a point or axis.
• Calculation: Moment = Force × Perpendicular distance from the point/axis to the line of action of the force
• Varignon’s theorem: The moment of a force about a point is equal to the sum of the moments of its components about the same point.
• Principle of moments for equilibrium: A body is in equilibrium if the sum of the moments of the forces acting on it about any point is zero. -Torque:
  • Definition: Torque is a measure of the twisting or rotational effect of a force.
  • Relationship to moment of force: Torque = Moment of force

2.Center of Gravity

References: NCERT Physics Class 11, Chapter 7: System of Particles and Rotational Motion

• Definition: Center of gravity of an object is the point where the entire weight of the object is considered to be concentrated.
• Methods to find the center of gravity:
  • For symmetrical objects, the center of gravity is at the geometrical center.
  • For irregular objects, the center of gravity can be determined by suspension method, balancing method, or by using the principle of moments.
• Properties and applications:
  • The weight of an object acts vertically downwards through its center of gravity.
  • The center of gravity is the point at which the resultant buoyant force acts on a floating object.
  • The stability of an object depends on its center of gravity.

3.Conditions for Equilibrium

References: NCERT Physics Class 11, Chapter 7: System of Particles and Rotational Motion

• Translational equilibrium: A body is in translational equilibrium if the net force acting on it is zero.
• Rotational equilibrium: A body is in rotational equilibrium if the net torque acting on it is zero.
• Equations of equilibrium:
• For a particle: ∑ F = 0
• For a rigid body: ∑ F = 0 and ∑ τ = 0
• Graphical representation: Free-body diagrams can be used to graphically represent forces and torques acting on a body.

4.Stability and Center of Mass

References: NCERT Physics Class 11, Chapter 7: System of Particles and Rotational Motion

• Stability:
• A body is stable if it returns to its equilibrium position after being slightly displaced.
• A body is unstable if it does not return to its equilibrium position after being slightly displaced.
• Center of mass:
  • The center of mass of a system of particles is the point at which the total mass can be considered to be concentrated.
  • For a uniform object, the center of mass is the same as the center of gravity.

5.Floating and Buoyancy

References: NCERT Physics Class 11, Chapter 8: Gravitation

• Archimedes’ principle:
• A body immersed in a fluid experiences an upthrust equal to the weight of the fluid displaced by the body.
• Buoyant force: The upthrust experienced by a body immersed in a fluid is called the buoyant force.
• Conditions for equilibrium of floating bodies:
  • The weight of the body is equal to the buoyant force.
  • The center of gravity of the body is vertically above the center of buoyancy.
• Metacenter:
• The metacenter of a floating object is the point where the line of action of the buoyant force intersects the vertical line through the center of gravity when the object is slightly tilted.
• The stability of a floating object depends on the position of the metacenter.

6.Application to Structures and Machines

References: NCERT Physics Class 11, Chapter 7: System of Particles and Rotational Motion; Physics Class 12, Chapter 9: Mechanical Properties of Solids

• Analysis of forces and moments:
  • Analyze forces and moments acting on structures like beams, levers, and pulleys.
• Calculation of reaction forces and internal forces:
  • Determine reaction forces at supports and internal forces within structures.
• Design and optimization:
• Use equilibrium principles to design and optimize structures and machines.

7.Problem-Solving Techniques

References: NCERT Physics Class 11, Chapter 7: System of Particles and Rotational Motion

• Free-body diagrams:
• Draw free-body diagrams to represent forces acting on a body or a system of particles.
• Algebraic methods:
• Use algebraic equations to solve equilibrium problems.
• Graphical methods:
• Use graphical methods (e.g., vector polygons, Mohr’s circle) to analyze forces and moments.
• Conservation laws:
• Apply conservation laws (e.g., conservation of energy) to solve equilibrium problems.
• Interpretation and analysis:
• Interpret and analyze equilibrium configurations and make predictions about the behavior of objects in equilibrium.