physics week 11 part 2 (ONLY SLIDES)

Topics Covered in Today's Class

  • Momentum

  • Impulse-Momentum Theorem

  • Momentum conservation

  • Center of mass

Conservation of Linear Momentum

Momentum of a System of Particles

  • Definition: The total momentum of a system is the sum of individual momentums.

  • Formula:
    {Total Momentum} (P) = p1 + p2 + \ldots + pN = \sum{k} p_k

  • Change in Momentum:
    \Delta P = \sum{k} \mathbf{F}{\text{ext on } k} \Delta t = \sum \mathbf{F}_{\text{ext}}

Law of Conservation of Momentum

  • Statement: The total momentum (P) of an isolated system is constant.

  • Implication: Interactions within the system do not change the system's total momentum.

  • Mathematical Formulation: \Delta P = 0 \Rightarrow P = \text{constant}

    • In an isolated system: \mathbf{F}_{\text{ext}} = 0

    • Initial and final momentum:
      P{i} = P{f}

  • Conclusion: The total momentum of the system after an event (e.g., a collision or explosion) is equal to the total momentum of the system before the event, although the momentum of each individual object may change during the event.

Impulse Approximation

  • Definition: During collisions and explosions, the duration is brief, and peak forces are significant. Therefore, other forces acting on particles, such as gravity, impart a comparatively insignificant impulse, which can be neglected.

  • Notation:

    • P_i = momentum immediately before the event

    • P_f = momentum immediately after the event

  • Under the approximation:
    Pi = Pf

Center of Mass

Definition

  • The center of mass is a useful concept when dealing with multiple objects or parts of a single object.

  • For an object with N parts and total mass M: \mathbf{r}{cm} = \frac{1}{M} \sum{j=1}^{N} mj \mathbf{r}j

    • Description: It can be thought of as the balancing point of the system.

Center of Mass & Momentum

  • If there are no external forces acting on the system, the momentum of the center of mass remains constant.

  • Assumption: If mass is constant, then the velocity of the center of mass is also constant.

Examples

Example 1: Space Junk

  • Context: A hazard of space travel includes debris from previous missions.

  • Description: There are thousands of detectable objects orbiting Earth, with an even greater number of small objects (like paint flakes).

  • Problem Calculation: Calculate the force exerted by a 0.100 mg chip of paint striking a spacecraft window at a speed of 4000 m/s, considering a collision duration of 60 nanoseconds.

Example 2: Firecracker

  • Context: A firecracker is launched vertically and explodes at the peak of its trajectory.

  • Details: If a 100 g firecracker explodes into two pieces, with a 72 g piece projected to the left at 20 m/s, the speed and direction of the other piece must be determined.

Reiteration of Example 2

  • Restates the scenario with emphasis on the outcome of the 100 g firecracker exploding into two pieces. The process for determining the speed and direction of the second piece is crucial for understanding conservation of momentum in explosive events.