Motion of System of Particles and Rigid Bodies

1. Center of Mass:

• Definition: The center of mass of a system of particles is the point where the entire mass of the system can be considered to be concentrated.
• Finding the center of mass: For regular bodies, the center of mass can be found using symmetry considerations. For irregular bodies, the center of mass can be found by integrating the mass distribution over the volume of the body. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.1)

2. Linear Momentum and Conservation of Momentum:

• Definition: Linear momentum of a particle is defined as the product of its mass and velocity.
• Impulse and momentum-impulse theorem: Impulse is defined as the product of force and time. The momentum-impulse theorem states that the net impulse acting on a particle is equal to the change in its momentum. (Refer to NCERT Physics Class 11, Chapter 6, Section 6.1 and 6.2)
• Conservation of momentum: The law of conservation of momentum states that the total momentum of a closed system remains constant, regardless of the internal interactions within the system. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.5)

3. Rotational Motion:

• Angular displacement, angular velocity, and angular acceleration: Angular displacement is the angle through which a body rotates. Angular velocity is the rate of change of angular displacement. Angular acceleration is the rate of change of angular velocity. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.6)
• Torque and the rotational analogue of Newton’s second law: Torque is the rotational analogue of force. The rotational analogue of Newton’s second law states that the net torque acting on a rigid body is equal to the rate of change of its angular momentum. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.7 and 7.8)
• Moment of inertia: Moment of inertia is a measure of the resistance of a rigid body to rotational motion. It depends on the mass distribution of the body. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.9)
• Parallel and perpendicular axis theorems: These theorems provide methods for calculating the moment of inertia of a rigid body about any axis, given its moment of inertia about a parallel or perpendicular axis. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.10)
• Rotational kinetic energy and work-energy theorem for rotational motion: Rotational kinetic energy is the energy possessed by a rotating rigid body. The work-energy theorem for rotational motion states that the net work done on a rigid body is equal to the change in its rotational kinetic energy. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.11 and 7.12)

4. Moment of Inertia:

• Definition and calculation of moment of inertia for simple objects: The moment of inertia of a rigid body about an axis is a measure of its resistance to rotation about that axis. It depends on the mass distribution of the body and the distance of the particles from the axis of rotation. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.9)
• Moment of inertia for composite objects: The moment of inertia of a composite object can be calculated by summing up the moments of inertia of its individual components. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.10)
• Parallel axis theorem and perpendicular axis theorem: These theorems provide methods for calculating the moment of inertia of a rigid body about any axis, given its moment of inertia about a parallel or perpendicular axis. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.10)

5. Angular Momentum and Conservation of Angular Momentum:

• Definition of angular momentum: Angular momentum of a particle is defined as the product of its moment of inertia and its angular velocity. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.13)
• Conservation of angular momentum: The law of conservation of angular momentum states that the total angular momentum of a closed system remains constant, regardless of the internal interactions within the system. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.15)
• Relationship between linear and angular momentum: The angular momentum of a system is related to its linear momentum by the formula L = r x p, where L is the angular momentum, r is the position vector from the point of rotation, and p is the linear momentum.

6. Equilibrium and Stability:

• Conditions for equilibrium of a rigid body: A rigid body is in equilibrium when the net force and the net torque acting on it are both zero. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.14)
• Potential energy and equilibrium: Potential energy can be used to determine the equilibrium positions of a rigid body. A rigid body is in equilibrium when it is in a position where its potential energy is minimum. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.14)
• Stability of equilibrium: Equilibrium can be stable, unstable, or neutral. Stable equilibrium occurs when a small displacement of the body from its equilibrium position results in a restoring force that brings the body back to equilibrium. Unstable equilibrium occurs when a small displacement of the body from its equilibrium position results in a force that further displaces the body from equilibrium. Neutral equilibrium occurs when a small displacement of the body from its equilibrium position results in no net force acting on the body. (Refer to NCERT Physics Class 11, Chapter 7, Section 7.14)
• Applications to simple mechanical systems: Equilibrium and stability concepts are applied in simple mechanical systems such as levers, pulleys, and inclined planes. (Refer to NCERT Physics Class 11, Chapter 8)

7. Simple Harmonic Motion:

• Definition and characteristics of simple harmonic motion (SHM): SHM is a periodic motion in which the restoring force is directly proportional to the displacement from the mean position. (Refer to NCERT Physics Class 11, Chapter 15)
• Equations of motion for SHM: The equations of motion for SHM are: x = A cos(ωt + φ), v = -Aω sin(ωt + φ), and a = -Aω² cos(ωt + φ), where A is the amplitude, ω is the angular