PHYSICS-C II: MECHANICS - Momentum of variable mass system

 PHYSICS-C II: MECHANICS

(Credits: Theory-04, Practicals-02)

Theory: 60 Lectures

Fundamentals of Dynamics: Inertial frames; Review of Newton’s Laws of Motion. Momentum of variable mass system: motion of rocket. Dynamics of a system of particles. Principle of conservation of momentum. Impulse. Determination of Centre of Mass of discrete and continuous objects having cylindrical and spherical symmetry (1-D, 2-D & 3-D).

Momentum of variable mass system

The concept of momentum in classical mechanics is typically associated with constant mass systems, where the mass of an object remains unchanged. However, in certain situations, such as rocket propulsion or the ejection of mass in various systems, the mass of an object can change over time. In such cases, we need to consider the momentum of a variable mass system.

 

Definition of Momentum for Variable Mass Systems:

In a variable mass system, the total momentum of the system is not only influenced by the motion of the object but also by the change in mass over time. The general expression for the total momentum (PP) of a variable mass system is given by:

P = m_sys⋅v_sys+∫([dm/dt]⋅v) dt

where:

• m_sys​ is the remaining mass of the system.
• v_sys​ is the velocity of the remaining mass.
• dm/dt is the rate of change of mass with respect to time.
• v is the velocity of the mass being expelled or added.

Derivation and Explanation:

1. Conservation of Linear Momentum:
• The principle of conservation of linear momentum states that the total linear momentum of a system of objects remains constant if no external forces act on the system. In a variable mass system, this principle is extended to account for changes in mass.
2. Rate of Change of Momentum:
• The rate of change of momentum of the system is equal to the net external force acting on the system according to Newton's second law (F_ext=dP/dt​).
3. Variable Mass System Equation:
• By applying the definition of momentum for variable mass systems, we arrive at the equation mentioned earlier. The first term on the right side represents the momentum of the remaining mass, while the second term accounts for the momentum associated with the changing mass.
4. Applications:
• This concept is crucial in understanding the motion of rockets. As a rocket expels mass through its exhaust, it gains forward momentum due to the expelled mass's backward momentum. This is described by the rocket equation.
5. Integral Interpretation:
• The integral in the equation represents the accumulated momentum over time due to the continuous expulsion or addition of mass. It is the result of integrating the product of the rate of change of mass and the velocity of the expelled or added mass with respect to time.

Key Points:

• Variable Mass System Context:
• This concept is applicable in scenarios where mass is continuously added or expelled, such as rocket propulsion or fluid dynamics problems involving changing mass.
• Rocket Motion:
• For rockets, the concept helps explain how the rocket's velocity changes as it expels mass through the rocket nozzle.
• Analytical Challenges:
• Analyzing variable mass systems can be more complex than constant mass systems due to the need to account for changing mass.

Understanding the momentum of variable mass systems is crucial for accurate predictions and engineering in situations where mass is not conserved. The concept provides a broader perspective on momentum conservation in dynamic systems with changing masses.