Notes from Toppers
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.