Conservation of Momentum and Center of Mass

Conservation of Momentum

Particle Momentum

The concept of momentum is fundamental in physics, describing the motion of a particle as the product of its mass and velocity. The impulse-momentum theorem provides a conservation law for particle motion.

• Impulse-Momentum Theorem: The change in momentum of a particle is equal to the impulse applied to it.

$\int \vec{F}_{\text{net}} dt = \int d\vec{p} = \Delta \vec{p}$

• Conservation Condition: When $\vec{F}_{\text{net}} = 0$, the particle's momentum remains constant.

• Extension to Systems: Momentum is also conserved for a system of particles, not just individual particles.

Momentum Conservation for a System of Objects

In many physical situations, we consider systems composed of multiple interacting objects. If the system is isolated (no significant external forces), its total momentum is conserved.

• Isolated System: A collection of objects with negligible interaction with the environment (e.g., solar system, atom, molecule).

• Conservation Law for Systems: The total momentum of the system changes only if external forces act on it.

$\int \vec{F}_{\text{ext}} dt = \int d\vec{p} = \Delta \vec{p} = \sum_i \Delta \vec{p}_i$

Internal and External Forces

Forces within a system can be classified as internal or external, depending on their origin.

• Internal Forces: Forces between objects within the system (e.g., gravitational force between two planets in the solar system).

• External Forces: Forces exerted on system objects by sources outside the system.

Example: If a system consists of objects A and B, the force A exerts on B is internal, while a force from an external object C on A or B is external.

The Momentum of a System

The total momentum of a system is the vector sum of the momenta of all its constituent objects.

• System Momentum:

$\vec{P} = \sum_{i=1}^N \vec{p}_i$

• Newton's Second Law for Object $i$:

$\vec{F}_i^{\text{net}} = m_i \vec{a}_i = \frac{d\vec{p}_i}{dt}$

• Newton's Second Law for the System:

$\sum_{i=1}^N \vec{F}_i^{\text{net}} = \sum_{i=1}^N \frac{d\vec{p}_i}{dt} = \frac{d\vec{P}}{dt}$

Cancellation of Internal Forces

Internal forces within a system always occur in equal and opposite pairs (Newton's Third Law), so their net effect on the system's total momentum cancels out.

• Sum of Forces in the System:

$\sum_{\text{system}} \vec{F} = \sum \vec{F}_{\text{int}} + \sum \vec{F}_{\text{ext}}$

• Internal forces sum to zero: $\sum \vec{F}_{\text{int}} = 0$

• Thus, $\sum_{\text{system}} \vec{F} = \sum \vec{F}_{\text{ext}}$

Conservation of Momentum

Only external forces can change the momentum of a system. If the sum of external forces is zero, the system's momentum is conserved.

• General Law:

$\frac{d\vec{P}}{dt} = \sum \vec{F}_{\text{ext}}$

• Isolated System: If $\sum \vec{F}_{\text{ext}} = 0$, then $\frac{d\vec{P}}{dt} = 0$ and momentum is conserved.

• Note: Energy and momentum are distinct; energy can change without affecting momentum.

Example: Two Carts on a Track

Consider two carts initially at rest. An internal spring pushes them apart. Since the system is isolated, momentum is conserved.

• Initial Momentum: $P_{\text{before}} = 0$

• Final Momentum: $P_{\text{after}} = m_1 v_1 + m_2 v_2 = 0$

• Relation: $m_1 v_1 = -m_2 v_2$

• Application: This principle is used in recoil problems and explosion analysis.

Momentum Conservation in Multiple Dimensions

Momentum Conservation in Three Dimensions

Momentum is a vector quantity, so conservation applies independently to each component (x, y, z).

• Component Equations:

$\frac{dP_x}{dt} = \sum F_{\text{ext},x}$ $\frac{dP_y}{dt} = \sum F_{\text{ext},y}$ $\frac{dP_z}{dt} = \sum F_{\text{ext},z}$

• Each direction of motion is independent.

Collisions in Two Dimensions

In two-dimensional collisions, momentum is conserved separately in both the x and y directions.

• Before Collision:

$P_{x,\text{before}} = m_1 v_{1,x,b} + m_2 v_{2,x,b}$ $P_{y,\text{before}} = m_1 v_{1,y,b} + m_2 v_{2,y,b}$

• After Collision:

$P_{x,\text{after}} = m_1 v_{1,x,a} + m_2 v_{2,x,a}$ $P_{y,\text{after}} = m_1 v_{1,y,a} + m_2 v_{2,y,a}$

• Momentum conservation in each direction allows us to solve two-dimensional problems as two separate one-dimensional problems.

Center of Mass

Definition and Calculation

The center of mass is a special point in a system of particles that moves as if all the system's mass were concentrated there and all external forces were applied at that point.

• Mathematical Definition:

$x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$ $y_{\text{cm}} = \frac{\sum m_i y_i}{\sum m_i}$

• Total Mass: $M = \sum_{i=1}^N m_i$

• The center of mass is the mass-weighted average position of all objects in the system.

Example: Two Objects

Given two objects with masses $m_1 = 4\ \text{kg}$ at $(2,6)$ and $m_2 = 2\ \text{kg}$ at $(8,2)$, the center of mass is calculated as:

$x_{\text{cm}} = \frac{1}{4+2}(4 \cdot 2 + 2 \cdot 8) = \frac{1}{6}(8 + 16) = 4$ $y_{\text{cm}} = \frac{1}{4+2}(4 \cdot 6 + 2 \cdot 2) = \frac{1}{6}(24 + 4) = 4.67$

• Interpretation: The center of mass lies closer to the heavier object.

Center-of-Mass Velocity and System Momentum

The velocity of the center of mass is related to the total momentum of the system:

$\vec{v}_{\text{cm}} = \frac{d\vec{r}_{\text{cm}}}{dt} = \frac{1}{M} \sum m_i \vec{v}_i$ $\vec{P} = M \vec{v}_{\text{cm}}$

• If there are no external forces, both the total momentum and the center-of-mass velocity remain constant.

Summary Table: Key Concepts in Momentum Conservation

Concept	Definition/Formula	Notes
Impulse-Momentum Theorem	$\int \vec{F}_{\text{net}} dt = \Delta \vec{p}$	Relates force and change in momentum
System Momentum	$\vec{P} = \sum \vec{p}_i$	Vector sum over all objects
Conservation Law	$\frac{d\vec{P}}{dt} = \sum \vec{F}_{\text{ext}}$	Momentum changes only by external forces
Center of Mass	$x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$	Mass-weighted average position
2-D Collisions	$m_1 v_{1,x,b} + m_2 v_{2,x,b} = m_1 v_{1,x,a} + m_2 v_{2,x,a}$ $m_1 v_{1,y,b} + m_2 v_{2,y,b} = m_1 v_{1,y,a} + m_2 v_{2,y,a}$	Momentum conserved in each direction

Take-Away Concepts

• Systems: Distinguish between internal and external forces.

• Momentum of a System: $\vec{P} = \sum \vec{p}_i$

• Newton's Second Law for a System: $\frac{d\vec{P}}{dt} = \sum \vec{F}_{\text{ext}}$

• Momentum Conservation: If $\sum \vec{F}_{\text{ext}} = 0$, then $\vec{P}$ is constant.

• Component Conservation: Momentum conservation applies to each component (x, y, z) separately.

• Center of Mass: $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$

• System Momentum and Center-of-Mass Velocity: $\vec{P} = M \vec{v}_{\text{cm}}$

Example Problem

Problem: Penguin A is at rest on a sled on frictionless ice. Penguin A walks toward penguin B on shore. What happens to the center of mass of the system (penguin A + sled)?

• Answer: The center of mass remains the same distance from the shore, since there are no external horizontal forces acting on the system.