Motion of Center of Mass, Relative Motion, and Reduced Mass

Note: The content below includes specific page references to the NCERT Physics textbooks for Class 11 and Class 12. These textbooks serve as the primary resource for the JEE exam.

Motion of Center of Mass

• Definition and properties of center of mass:

  • The center of mass of a system of particles is a unique point where the total mass of the system can be considered to be concentrated (NCERT Class 11, Chapter 9, Page 143).
  • The center of mass is independent of the frame of reference (NCERT Class 12, Chapter 3, Page 55).

• Calculating the center of mass:

  • For simple objects with uniform density, the center of mass is located at the geometric center (NCERT Class 11, Chapter 9, Page 145).
  • For complex objects, the center of mass can be determined by integration or by the “method of moments” (NCERT Class 11, Chapter 9, Pages 147-148).

• Equations of motion for the center of mass:

  • The center of mass of a system of particles moves as if all of the mass were concentrated at the center of mass and subjected to the net force acting on the system (NCERT Class 11, Chapter 9, Page 150).
  • (\sum \overrightarrow{F} = M \overrightarrow{a}{CM}), where (\overrightarrow{F}) is the net force, (M) is the total mass of the system, and (\overrightarrow{a}{CM}) is the acceleration of the center of mass.

• Conservation of momentum:

  • The total momentum of a closed system remains constant, even if the internal forces change the velocities of individual objects within the system (NCERT Class 11, Chapter 9, Page 97).
  • (\overrightarrow{P}_{total} = \sum m_i\overrightarrow{v}_i = \text{constant}), where (m_i) is the mass of the (i)-th particle and (\overrightarrow{v}_i) is its velocity.

Relative Motion

• Definition and examples of relative motion:

  • Relative motion is the motion of an object with respect to another object that is considered stationary (NCERT Class 11, Chapter 8, Page 119).
  • Examples include a person walking on a moving walkway or a passenger in a moving car observing the scenery outside (NCERT Class 11, Chapter 8, Pages 122-123).

• Relative velocity and acceleration:

  • The relative velocity of two objects is the difference between their velocities (NCERT Class 11, Chapter 8, Page 123).
  • The relative acceleration of two objects is the difference between their accelerations (NCERT Class 11, Chapter 8, Page 124).

• Equations for relative velocity and acceleration:

  • (\overrightarrow{v}_{AB} = \overrightarrow{v}_A - \overrightarrow{v}B), where (\overrightarrow{v}{AB}) is the relative velocity of object (A) with respect to object (B), and (\overrightarrow{v}_A) and (\overrightarrow{v}_B) are the velocities of objects (A) and (B), respectively.
  • (\overrightarrow{a}_{AB} = \overrightarrow{a}_A - \overrightarrow{a}B), where (\overrightarrow{a}{AB}) is the relative acceleration of object (A) with respect to object (B), and (\overrightarrow{a}_A) and (\overrightarrow{a}_B) are the accelerations of objects (A) and (B), respectively.

• Applications of relative motion:

  • Determining the motion of objects on rotating platforms (NCERT Class 11, Chapter 8, Pages 126-128).
  • Analyzing the motion of projectiles (NCERT Class 11, Chapter 3, Pages 33-34).

Reduced Mass

• Introduction and definition of reduced mass:

  • The reduced mass of a two-particle system is a concept used to simplify the analysis of motion in two-body problems (NCERT Class 12, Chapter 4, Page 83).
  • It is defined as: (m_{red} = \frac{m_1 m_2}{m_1 + m_2}), where (m_1) and (m_2) are the masses of the two particles.

• Applications of reduced mass:

  • Simplifying the analysis of the motion of a satellite orbiting a planet (NCERT Class 12, Chapter 4, Page 84).
  • Understanding the motion of a projectile in a gravitational field (NCERT Class 12, Chapter 3, Pages 51-52).

Problem-Solving Techniques

• Identify relevant concepts:

  • Analyze the given information and identify the key concepts related to motion of center of mass, relative motion, and/or reduced mass.

• Apply physical principles:

  • Apply the relevant principles and equations, such as conservation of momentum, equations of motion, and the concept of relative velocity/acceleration.

• Solve for unknowns:

  • Solve the mathematical equations to determine the required quantities, such as velocity, acceleration, or position.

• Practice, practice, practice!:

  • Regular practice with a variety of problems will help reinforce understanding and improve problem-solving skills.