### 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.