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:
  ${Total Momentum} (P) = p<em>1 + p</em>2 + \ldots + p<em>N = \sum</em>{k} p_k$

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

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:
    $P<em>{i} = P</em>{f}$

• 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:
  $P<em>i = P</em>f$

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: $\mathbf{r}<em>{cm} = \frac{1}{M} \sum</em>{j=1}^{N} m<em>j \mathbf{r}</em>j$

  • 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.