Lecture 4.1

Overview

• Lecture Title: Physics

• Lecture No: 4

• Lecturer: Nurakhmetova S.

• Department: RET Department, International IT University

Topics Covered

• Linear Momentum and Collisions

  • Linear Momentum

  • Isolated System (Momentum)

  • Non-isolated System (Momentum)

  • Collisions in One Dimension

  • Collisions in Two Dimensions

  • Center of Mass

  • Systems of Many Particles

Linear Momentum

• Definition: Linear momentum (p) is defined as the product of mass (m) and velocity (V).

• Formula: p = mV

• Characteristics:

  • Momentum is a vector quantity, meaning it has both magnitude and direction.

  • SI units are kg·m/s.

  • Dimensions: ML/T.

• Component Form:

  • p_x = m * v_x

  • p_y = m * v_y

  • p_z = m * v_z

• Significance: Momentum measures how difficult it is to stop or change the direction of a moving object.

Newton’s Laws and Momentum

• Newton’s Second Law: Relates momentum to the resultant force acting on an object with constant mass.

Conservation of Linear Momentum

• Principle: In an isolated system, the total momentum remains constant during interactions. The system's total momentum equals its initial momentum.

• Mathematical Expression:

  • p_initial = p_final

• Component Form:

  • p_{ix} = p_{fx},

  • p_{iy} = p_{fy},

  • p_{iz} = p_{fz}

• Application: Valid for any number of particles.

Impulse and Momentum

• Impulse (J): The change in momentum due to a force applied over time.

• Impulse-Momentum Theorem:

  • Impulse = Change in momentum (J = Δp)

• SI Unit of Impulse: N·s

• Magnitude: Equal to the area under the force-time curve.

Types of Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved.

  • Occurs at a microscopic level or approximately in macroscopic interactions.

• Inelastic Collision: Momentum is conserved, kinetic energy is not. If objects stick together after collision, it's perfectly inelastic.

• Momentum Conservation: Always applies in collisions, regardless of type.

Center of Mass

• Definition: A special point in a system where mass can be considered to be concentrated.

• Motion: Follows the resultant external force acting on the system.

• Acceleration: Given by ΣF_ext/M.

• Importance of Center of Mass: Allows description of motion of a mechanical system as if all mass were at a single point.

Differences between Center of Mass and Center of Gravity

• Center of Mass (CM): Location where mass distribution is equal in all directions; unaffected by gravity.

• Center of Gravity (CG): Where the weight distribution equals out; affected by gravitational field.

• In uniform gravitational fields, CG coincides with CM, but can differ in non-uniform fields.

Motion of a System of Particles

• Described in terms of the center of mass’s velocity and acceleration.

• Total linear momentum = Total mass × Velocity of CM.

• Acceleration determined by differentiating velocity and related to external forces acting on the system.

Key Formulae

• Momentum: p = mV

• Impulse-Momentum Theorem: J = Δp

• Center of Mass Acceleration: a_cm = ΣF_ext/M

• Conservation of Momentum (2D): m1V1i + m2V2i = m1V1f + m2V2f (for both x and y directions).

Conclusion

Understanding momentum and collisions is critical in physics, as they demonstrate fundamental principles such as conservation laws and the relationship between forces, mass, and acceleration.