Mr. Lowrie's Science Site
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  • Environmental Systems
    • First Week Stuff
    • Key Terms Glossary
    • Unit 1: Introduction to Environmental Science >
      • Chapter 1: Science and the Environment >
        • Section 1: Understanding Our Environment
        • Section 2: The Environment and Society
      • Chapter 2: Tools of Environmental Science >
        • Section 1: Scientific Methods
        • Section 2: Statistics and Models
        • Section 3: Making Informed Decisions
      • Chapter 3: The Dynamic Earth >
        • Section 1: The Geosphere
        • Section 2: The Atmosphere
        • Section 3: The Hydrosphere and Biosphere
    • Unit 2: Ecology >
      • Chapter 4: The Organization of Life >
        • Section 1: Ecosystems: Everything is Connected
        • Section 2: Evolution
        • Section 3: The Diversity of Living Things
      • Chapter 5: How Ecosystems Work >
        • Section 1: Energy Flow in Ecosystems
        • Section 2: The Cycling of Materials
        • Section 3: How Ecosystems Change
      • Chapter 6: Biomes >
        • Section 1: What is a Biome?
        • Section 2: Forest Biomes
        • Section 3: Grassland, Desert, and Tundra Biomes
      • Chapter 7: Aquatic Ecosystems >
        • Section 1: Freshwater Ecosystems
        • Section 2: Marine Ecosystems
    • Unit 3: Populations >
      • Chapter 8: Understanding Populations >
        • Section 1: How Populations Change in Size
        • Section 2: How Species Interact with Each Other
      • Chapter 9: The Human Population >
        • Section 1: Studying Human Populations
        • Section 2: Changing Population Trends
      • Chapter 10: Biodiversity >
        • Section 1: What is Biodiversity?
        • Section 2: Biodiversity at Risk
        • Section 3: The Future of Biodiversity
    • Unit 4: Water, Air, and Land >
      • Chapter 11: Water >
        • Section 1: Water Resources
        • Section 2: Water Use and Management
        • Section 3: Water Pollution
      • Chapter 12: Air >
        • Section 1: What Causes Air Pollution?
        • Section 2: Air, Noise, and Light Pollution
        • Section 3: Acid Precipitation
      • Chapter 13: Atmosphere and Climate Change >
        • Section 1: Climate and Climate Change
        • Section 2: The Ozone Shield
        • Section 3: Global Warming
      • Chapter 14: Land >
        • Section 1: How We Use Land
        • Section 2: Urban Land Use
        • Section 3: Land Management and Conservation
      • Chapter 15: Food and Agriculture >
        • Section 1: Feeding the World
        • Section 2: Crops and Soil
        • Section 3: Animals and Agriculture
    • Unit 5: Mineral and Energy Resources >
      • Chapter 16: Mining and Mineral Resources >
        • Section 1: Minerals and Mineral Resources
        • Section 2: Mineral Exploration and Mining
        • Section 3: Mining Regulations and Mine Reclamation
      • Chapter 17: Nonrenewable Energy >
        • Section 1: Energy Resources and Fossil Fuels
        • Section 2: Nuclear Energy
      • Chapter 18: Renewable Energy >
        • Section 1: Renewable Energy Today
        • Section 2: Alternative Energy and Conservation
      • Chapter 19: Waste >
        • Section 1: Solid Waste
        • Section 2: Reducing Solid Waste
        • Section 3: Hazardous Wastes
    • Unit 6: Our Health and Future >
      • Chapter 20: The Environment and Human Health >
        • Section 1: Pollution and Human Health
        • Section 2: Biological Hazards
      • Chapter 21: Economics, Policy, and the Future >
        • Section 1: Economics and International Cooperation
        • Section 2: Environmental Policies in the United States
        • Section 3: The Importance of the Individual
  • AP Environmental Science
    • First Week Stuff
    • Unit I: Humans and Sustainability: An Overview >
      • Chapter 1: Environmental Problems, Their Causes, and Sustainability
    • Unit II: Science, Ecological Principles, and Sustainability >
      • Chapter 2: Science, Matter, Energy, and Systems
      • Chapter 3: Ecosystems: What are They and How Do They Work?
      • Chapter 4: Biodiversity and Evolution
      • Chapter 5: Biodiversity, Species Interactions, and Population Control
      • Chapter 6: The Human Population and Its Impact
      • Chapter 7: Climate and Biodiversity
      • Chapter 8: Aquatic Biodiversity
    • Unit III: Sustaining Biodiversity >
      • Chapter 9: Sustaining Biodiversity: Saving Species and Ecosystem Services
      • Chapter 10: Sustaining Terrestrial Biodiversity: Saving Ecosystems and Ecosystem Services
      • Chapter 11: Sustaining Aquatic Biodiversity and Ecosystem Services
    • Unit IV: Sustaining Natural Resources >
      • Chapter 12: Food Production and the Environment
      • Chapter 13: Water Resources
      • Chapter 14: Nonrenewable Mineral Resources
      • Chapter 15: Nonrenewable Energy
      • Chapter 16: Energy Efficiency and Renewable Energy
    • Unit V: Sustaining Environmental Quality >
      • Chapter 17: Environmental Hazards and Human Health
      • Chapter 18: Air Pollution
      • Chapter 19: Climate Disruption
      • Chapter 20: Water Pollution
      • Chapter 21: Solid and Hazardous Waste
      • Chapter 22: Urbanization and Sustainability
    • Unit VI: Sustaining Human Societies >
      • Chapter 23: Economics, Environment, and Sustainability
      • Chapter 24: Politics, Environment, and Sustainability
      • Chapter 25: Environmental Worldviews, Ethics, and Sustainability
  • Chemistry
    • First Week Stuff
    • Matter

Center of Mass and Linear Momentum


Center of Mass:  The center of mass of a system of n particles is defined to be the point whose coordinates are given by:
Picture
where M is the total mass of the system.

Newton's second Law for a System of Particles:  The motion of the center of mass of any system of particles is governed by Newton's second law for a system of particles, which is:
Picture
Here F(sub)net is the net force of all the external forces acting on the system, M is the total mass of the system, and a(sub)com is the acceleration of the system's center of mass.

Linear Momentum and Newton's Second Law:  For a single particle, we define a quantity p, called its linear momentum as:
Picture
and can write Newton's second law in terms of this momentum:
Picture
For a system of particles these relations become:
Picture

Collision and Impulse:  Applying Newton's second law in momentum form to a particle-like body involved in a collision leads to the impulse-linear momentum theorem:
Picture
Where final momentum - original momentum is the change in the body's linear momentum and J is the impulse due to the force F(t) exerted on the body by the other body in the collision:
Picture
If F(sub)avg is the average magnitude of F(t) during the collision and delta t is the duration of the collision, then for one-dimensional motion:
Picture
When a steady stream of bodies, each with mass m and speed v, collides with a body whose position is fixed, the average force on the fixed body is:
Picture
where n/delta t is the rate at which the bodies collide with the fixed body, and delta v is the change in velocity of each colliding body.  This average force can also be written as:
Picture
where delta m/delta t is the rate at which mass collides with the fixed body.  In the previous two equations, delta v = -v if the bodies stop upon impact and delta v = -2v is they bounce directly backward with no change in their speed.

Conservation of Linear Momentum:  If a system is isolated so that no net external force acts on it, the linear momentum P of the system remains constant:
Picture
this can also be written as:
Picture
where the subscripts refer to the values of P at some original time and at a later time.  The two statements above are equivalent statements of the law of conservation of linear momentum.

Inelastic Collision in One Dimension:  In an inelastic collision of two bodies, the kinetic energy of the two-body system is not conserved.  If the system is closed and isolated, the total linear momentum of the system must be conserved, which we can write in vector form as:
Picture
where subscripts o and f refer to values just before and just after the collision, respectively(symbols for single objects are lowercase, symbols for systems of particles are capital letters, the above is lowercase, further down is capital). 
If the motion of the bodies is along a single axis, the collision is one-dimensional and we can rewrite the previous equation in terms of velocity components along that axis:
Picture
If the bodies stick together, the collision is completely inelastic and the bodies have the same final velocity V (because they are stuck together).

Motion of the Center of Mass:  The center of mass of a closed, isolated system of two colliding bodies is not affected by a collision.  In particular, the velocity v(sub)com of the center of mass cannot be changed by the collision.

Elastic Collisions in One Dimension:  An elastic collision is a special type of collision in which the kinetic energy of a system of colliding bodies is conserved.  If the system is closed and isolated, its linear momentum is also conserved.  For a one-dimensional collision in which body 2 is a target and body 1 is an incoming projectile, conservation of kinetic energy and linear momentum yield the following expressions for the velocities immediately after the collision:
Picture

Collisions in Two Dimensions:  If two bodies collide and their motion is not along a single axis (the collision is not head-on), the collision is two-dimensional.  If the two body system is closed and isolated, the law of conservation of momentum applies to the collision and can be written as:
Picture
In component form, the law gives two equations that describe the collision (one equation for each of the two dimensions).  If the collision is also elastic (a special case), the conservation of kinetic energy during the collision gives a third equation:
Picture

Variable-Mass Systems:  In the absence of external forces a rocket accelerates at an instantaneous rate given by:
Picture
in which M is the rocket's instantaneous mass (including unexpended fuel), R is the fuel consumption rate, and v(sub)rel is the fuel's exhaust speed relative to the rocket.  The term Rv(sub)rel is the thrust of the rocket engine.  For a rocket with constant R and v(sub)rel, whose speed changes from v(sub)0 to v(sub)f when its mass changes from M(sub)0 to M(sub)f (ln is natural log):
Picture

Files:
APCM Chapter 9 Presentation
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