Principle of Conservation of Linear Momentum

• This topic introduces the Principle of Conservation of Linear Momentum, which is a fundamental concept in physics.
• It is directly based on Newton's second law of motion, which relates force to the change in momentum.
• The principle states that if there is no net force acting on a system, its total linear momentum remains unchanged.
• This concept is essential for analyzing interactions like collisions and explosions in a simplified manner.

“The total momentum of an isolated system is conserved during any interaction.”

The process involves applying Newton's second law to individual bodies or systems using Free Body Diagrams (FBDs).

Fig 4.2 (a): System for illustration of free body diagram.

Steps for using Free Body Diagrams:

• Define System: Identify the specific body or a set of bodies (system) you are analyzing.
• Draw FBD: Create a Free Body Diagram, which shows all the individual forces acting on only one body at a time (and its acceleration).
• Identify Agencies: Identify all agencies exerting forces on that mass (e.g., Earth for gravity, strings for tension, contact surfaces for normal force/friction).
• Resolve Forces: Resolve forces into relevant perpendicular components (e.g., horizontal and vertical).

Free body diagram for a 2 kg mass.

Free body diagram for a 5 kg mass.

• Apply Newton's Second Law (\[\vec F\] = m\[\vec a\]):
  Write the force equation along the direction of motion: Resultant Force = mass × acceleration.
  For directions without motion, the resultant force is zero (forces must cancel).
• Simultaneous Solution: Write similar equations for all other masses/pulleys, and solve the simultaneous equations to find necessary quantities (e.g., tension and acceleration).

• It states that if the resultant force \[\vec F\] = 0, then the rate of change of linear momentum \[\frac{dp}{dt}= 0.\]
• This means the linear momentum \[\vec p\] is constant (or conserved).
• It simplifies the analysis of interactions by treating the interacting bodies as a single system.
• It is the fundamental principle used to analyse collisions and explosions.

Calculate the acceleration (a) and tension (T) for an Atwood machine with masses m₁ and m₂ (m₂ > m₁).

Given: m₂ > m₁. m₂ moves down, m₁ moves up.

Step 1: Write Free Body Equations (from source Method II)

• For m₁ (upward motion): T - m₁g = m₁a
• For m₂ (downward motion): m₂g - T = m₂a

Step 2: Solve for Acceleration (a)

• Add the two equations: (T − m₁​g) + (m_2​g − T) = m₁​a + m_2​am_2​g − m₁​g = a(m₁ ​+ m₂​) g(m₂​ − m₁​) = a(m₁​ + m_2​)
• Rearrange to find a:
  a = \[\left(\frac{m_2-m_1}{m_1+m_2}\right)g\]

Step 3: Solve for Tension (T)

• Using the equation for m₂: T = m₂g - m₂a = m₂(g - a)
• Substitute the expression for a:
  T = \[\left(\frac{2m_1m_2}{m_1+m_2}\right)g\]

• Rocket Propulsion: A rocket expels hot gases (mass) backward at high speed (momentum). The total momentum of the rocket and the gas is conserved. Since the gas gains backward momentum, the rocket gains forward momentum.
• Recoil of a Gun: When a bullet is fired forward, the gun is pushed backward. The initial total momentum (zero) is conserved; the forward momentum of the bullet is balanced by the backward momentum (recoil) of the gun.
• Collision between Billiard Balls: When two billiard balls collide, the vector sum of their momenta just before the collision is equal to the vector sum of their momenta just after the collision (assuming no significant external forces like friction).