Chapter 9: Momentum and Angular Momentum – Study Notes

Momentum

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. This principle is fundamental in analyzing interactions such as collisions and explosions.

• Isolated System: A system where the net external force is zero, so momentum is conserved.

• Impulse Approximation: For very short, intense interactions, external forces can be ignored, and momentum is conserved during the interaction.

• Mathematical Formulation: For a system of objects, the total momentum before an interaction equals the total momentum after: $m_1 v_{1i} + m_2 v_{2i} + \cdots = m_1 v_{1f} + m_2 v_{2f} + \cdots$

• Problem-Solving Steps:

  1. Strategize: Define the system and check for isolation.

  2. Prepare: Draw before-and-after diagrams, define symbols, and list knowns/unknowns.

  3. Solve: Apply conservation of momentum equations.

  4. Assess: Check units and reasonableness of the answer.

Explosions

An explosion is an event where the particles of a system move apart after a brief, intense interaction. Explosions are the opposite of collisions, and if the system is isolated, the total momentum is conserved.

• Internal Forces: The forces causing the explosion are internal, so the total system momentum remains unchanged.

• Example: A rocket expelling exhaust gases demonstrates conservation of momentum: the rocket moves forward as gases are ejected backward.

Inelastic Collisions

In a perfectly inelastic collision, two objects stick together and move with a common final velocity. Momentum is conserved, but kinetic energy is not.

• Examples: Clay hitting the floor, a bullet embedding in wood.

• Equation: $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f$

Momentum and Collisions in Two Dimensions

When collisions occur in two dimensions, momentum must be conserved in each component (x and y directions) separately.

• Component Equations:

  • $p_{1x,i} + p_{2x,i} + \cdots = p_{1x,f} + p_{2x,f} + \cdots$

  • $p_{1y,i} + p_{2y,i} + \cdots = p_{1y,f} + p_{2y,f} + \cdots$

• Application: The total vector momentum is conserved, even though individual momenta may change.

Example: Peregrine Falcon Strike (Two-Dimensional Collision)

This example analyzes a perfectly inelastic collision between a falcon and a pigeon, using conservation of momentum in two dimensions.

• Given: Falcon mass = 0.80 kg, speed = 18 m/s at 45°; Pigeon mass = 0.36 kg, speed = 9.0 m/s (horizontal).

• Find: Speed and direction of the combined mass after collision.

• Solution Steps:

  1. Break initial velocities into x and y components.

  2. Apply conservation of momentum in both directions.

  3. Find final velocity magnitude and direction using Pythagorean theorem and trigonometry.

• Key Equations:

  • $P_{x,i} = m_F (-v_F \cos \theta) + m_P (-v_P)$

  • $P_{y,i} = m_F (-v_F \sin \theta)$

  • $v_f = \sqrt{v_{x,f}^2 + v_{y,f}^2}$

  • $\tan \alpha = \frac{v_{y,f}}{v_{x,f}}$

Angular Momentum

Definition and Properties

Angular momentum is the rotational analog of linear momentum. It is conserved in the absence of external torques and is crucial for understanding rotational motion.

• Angular Momentum (L): $L = I \omega$ where $I$ is the moment of inertia and $\omega$ is the angular velocity.

• Comparison to Linear Momentum:

  Rotational Dynamics	Linear Dynamics
  Torque ($\tau_{net}$)	Force ($F_{net}$)
  Moment of inertia ($I$)	Mass ($m$)
  Angular velocity ($\omega$)	Velocity ($v$)
  Angular momentum ($L = I \omega$)	Linear momentum ($p = mv$)

Conservation of Angular Momentum

If the net external torque on a system is zero, the total angular momentum remains constant. This principle is analogous to the conservation of linear momentum.

• Mathematical Statement: $\tau_{net} \Delta t = \Delta L$

• Law of Conservation: If $\tau_{net} = 0$, then $L_f = L_i$

• System Application: The total angular momentum is the sum over all objects in the system.

• Equation for Multiple Objects: $$(I_1 \omega_1)_f + (I_2 \omega_2)_f + \cdots = (I_1 \omega_1)_i + (I_2 \omega_2)_i + \cdots$$

Varying Moment of Inertia

The moment of inertia of a system can change if the distribution of mass changes, even if the system is isolated. This leads to changes in angular velocity to conserve angular momentum.

• Example: Figure skaters spin faster by pulling their arms in, reducing their moment of inertia.

• Equation: $I_i \omega_i = I_f \omega_f$

Example: Period of a Merry-Go-Round

This example demonstrates conservation of angular momentum when a person moves from the center to the edge of a rotating merry-go-round.

• Given: Joey (36 kg) stands on a 200 kg merry-go-round (radius 2.0 m), initially rotating once every 2.5 s.

• Find: The new period after Joey walks to the edge.

• Solution Steps:

  1. Model the merry-go-round as a uniform disk: $I_{disk} = \frac{1}{2}MR^2$

  2. Joey's moment of inertia is $0$at the center,$mR^2$ at the edge.

  3. Apply conservation: $I_i \omega_i = I_f \omega_f$

  4. Relate angular velocity to period: $\omega = \frac{2\pi}{T}$

• Result: The period increases as Joey moves outward, since the moment of inertia increases and angular velocity decreases to conserve angular momentum.