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Units of measurement used to describe force, energy, and power.

• Foundational info
  • d = displacement = x --> m
    • d = distance. d is a vector but d is a scalar.
  • t = time --> s
  • m = mass --> Kg
    • I = moment of inertia = angular mass = k * m * r² --> Kg * m². Angular mass is the rotational inertia of a rigid body. k is a dimensionless constant called the inertial constant and it varies with different object. EG: k=1 for a thin ring or cylinder, k=2/5 for a solid sphere. Angular mass is a rotational analogy to mass.
  • θ = angle --> rad
  • ƒ = frequency --> Hz = s^-1
    • T = period --> s
    • ƒ = 1 / T
  • r = circumference / (2*pi) = diameter / 2 = radius --> m. A line segment from the center to the boundary of a circle.
  • rad = 360 degrees / (2*pi) = circumference / 2 * pi = radian. If a circle has r=1, then an angle of 1 rad has an arc equivalent to the radius.
• Scalars and vectors
  • |a| = sqrt(a₁² + a₂² + a₃²) = magnitude. A non-negative real number.
  • a = scalar. A magnitude multiplied with a unit of measure.
  • a = AB = a₁*(1,0,0) + a₂*(0,1,0) + a₃*(0,0,1) = a₁*i + a₂*j + a₃*k = a₁*e₁ + a₂*e₂ + a₃*e₃ = vector. A scalar with direction.
    • Unbound vectors are equal if they have the same magnitude and direction.
    • Bound vectors are equal if they have the same base point, magnitude and direction.
  • a + b = (a₁+b₁)*i + (a₂+b₂)*j + (a₃+b₃)*k = vector addition
  • a · b = a*b*cos θ = dot product = inner product = scalar product. θ is the angle between the two vectors. This yields a scalar.
  • a × b = -b × a = a * b * sin θ * n = cross product = area of the parallelogram formed by a and b. θ is the angle between the vectors. n is the unit vector perpendicular to the two vectors. This follows the right hand rule, so a × b is 180 degrees different from b × a. This yields a "pseudo-vector".
• d and derivatives
  • d = distance = x --> m
    • d = distance. d is a vector but d is a scalar.
  • v = velocity = translational velocity =  dx/dt --> m * s^-1
    • s = |v| = speed.
    • ω = angular velocity = dθ/dt = v / r --> rad * s^-1. Angular velocity is a pseudo-vector (following th right-hand-rule) that measures how fast an object rotates about an axis. Angular velocity is a rotational analogy to velocity.
      • ω = |ω| = angular speed = angular frequency.
    • The five equations of motion apply to bodies in linear motion and uniform acceleration (a=0).
      1. v_f = v_i + a*Δt. Where v_f=final velocity; v_i=initial velocity, a=constant acceleration, Δt=time between initial and final states.
      2. d = ½*(v_i + v_f)*Δt. Where d = distance between initial and final states.
      3. d = v_i*Δt + ½*a*Δt².
      4. (v_f)² = (v_i)² + 2*a*d
      5. d = v_f*Δt - ½*a*Δt².
  • a = acceleration = dv/dt = d²x/dt² --> m * s^-2
    • α = angular acceleration = -ω² * r --> m * s^-2. Angular acceleration is a rotational analogy to acceleration.
  • j = jerk = jolt = surge = da/dt = = d³x/dt³ --> m * s^-3
• p and derivatives
  • p = momentum = m*v --> Kg * m * s^-1
    • I = impulse = ∫F dt --> Kg * m * s^-1
      • I = FΔt if force constant.
      • I = Δp
    • L = angular momentum = I * ω --> Kg * m² * s^-1. Angular momentum is a rotational analogy to momentum.
  • F = force = dp/dt = d(m*v)/dt = m*a --> N = Kg * m * s^-2. Force: An external cause for potential change in a physical system. An influence which can cause a mass to accelerate. An object at rest frequently has forces working on it where the net force is zero.
    • τ = torque = dL/dt = d(I * ω)/dt = I * α = F * d --> N * m. Torque is a pseudo-vector (following the right-hand rule) that measures the tendency to rotate an object about an axis. Torque is synonymous with moment or moment of force, but the former is in the vernacular and engineering, while the latter is in physics. Torque is a rotational analogy to force —even though the units are that of energy!
      • A couple is a pair of equal torques in opposite directions. The net effect is a "moment", i.e. a turning (or prevention of turning) in place. Contrast turning a bolt with an Allen/hex/L-wrench versus with a T-wrench.
    • The four fundamental forces of nature:
      • strong nuclear force
      • weak nuclear force
      • electromagnetic force
      • gravitational force
    • F = centripetal force = -m * ω² * r --> N = Kg * m * s^-2. Centripetal force is the external force required to make a body follow a circular path at constant speed. EG: The gravitational force between the Sun and the Earth acts as a centripetal force inwards (towards the Sun) which keeps the Earth in orbit,
    • Conservative forces
      • Coulomb's force = force between electrical charges
      • Gravitational force = force between masses
      • Magnetic force
      • Spring force
    • Nonconservative
      • Drag force
      • Frictional force
    • Centrifugal force
    • Impact force
• E and derivatives
  • E = energy = F*d --> J = N * m = Kg * m² * s^-2 = 10⁷ erg
    • Work = ∫F · d
    • Work = F · d = F * d * cos φ
    • Energy is the ability to do work, i.e. apply force over a distance.
    • Energy is conserved. This is usually expressed as the First Law of Thermodynamics. The Second Law is roughly that entropy increases (and denotes the vector of time). The Third Law is roughly that entropy approaches a constant when absolute zero is approached.
    • Electrical. This is often measured in KW * h = 3.6 MJ.
    • Chemical
    • Radiant
    • Nuclear
    • Heat
    • Internal
    • Kinetic = ½ * m * v² = E_k
    • Potential = mass * Earth's gravitational acceleration * height = mass * 9.80665 m * s^-2 * height = m * g * h = U_g
    • Mechanical. Mechanical Energy is conserved: (E_k + E_p)₁ = (E_k + E_p)₂
    • Gravitational Potential = -G * m * M / r. Where G = gravitational constant = 6.6742 * 10^-11 * N * m² * Kg^-2
    • Nuclear Potential = ½ * m * speed of light squared = ½ * m * c²
  • P = power = dE/dt --> W = J / s = N * m * s^-1 = Kg * m² * s^-3. Power is the rate at which work/energy is performed/used. Thus the same work if done quickly takes more power.
    • P = τ × ω --> N * m * s^-1
    • Mild exercise is around 90 W, while vigorous exercise can be 900 W. As of 2001, the world uses around 13 TW, while the U.S. uses around 3.3 TW. See also Orders of magnitude (power) [W].

Miscellany

• A BTU (British Thermal Unit) is a unit of energy in the U.S. System. A BTU is the amount of heat it takes to raise 1 pound of water by 1 Fahrenheit, but since different authorities differ over what temperature range to set the standard for this the value of a BTU varies by about 0.5%. What a lousy unit of measure! 1 BTU ~ 1 match.
• An eV (electron volt) is a unit of energy equivalent to the 1 V * the electron charge of an electron. 1 eV = 1.602 176 53 (14)×10⁻¹⁹ J. Since mass and energy are equivalent (via E = m*c ²), the eV is commonly used to express mass in particle physics. EG: The mass of 1 proton = 1.672 621 71(29) × 10⁻²⁷ Kg = 938.272 029(80) MeV/c².
• Mechanical Advantage (MA) is the factor by which a "machine" multiplies the force put into it, i.e. roughly F _output / F _input. Since energy is conserved, the "trick" is usually to increase the distance since the same work is done, i.e. since W _{no MA} = W _MA, then (F*d) _{no MA} = (F _{smaller input} * d _larger ) _MA. The simple machines:
  • Lever. MA = d _{input/effort arm} / d _{output/load arm}. This is largely a matter of equivalent torques. Note that sometimes more output length/speed is desired instead of more output force. One mnemonic device is "FLEx", since Fulcrum, Load, and Effort are between the others in that order for the three classes of levers.
    1. First-class levers. The fulcrum is between the forces.
       • If the fulcrum is centered, then the movement will be in the direction of the greater force. EGs:
         • A see-saw. The more balanced the forces, the more easily see-saw moves.
       • If the fulcrum is closer to the load, then a small effort can move a large distance in order to move a large load a small distance. EGs:
         • Crowbar prying a nail.
       • If the fulcrum is closer to the effort, then a great effort can move a small distance in order to move a small load a great distance.. EGs:
         • Trebuchets drop great weights/effort suddenly to hurl objects/loads.
    2. Second-class levers. The load is between the effort and the fulcrum. Usually a small effort can move a large distance in order to move a large load a small distance. EGs:
       • A wheelbarrow.
       • A nut cracker has two second-class levers.
       • The part of a nail clipper pressed by the thumb is a second-class lever.
    3. Third-class levers. The effort is between the load and the fulcrum. Usually an effort is applied a small distance in order to move the load a large distance. EGs:
       • In rapier, the cavazione is a fast third-class lever motion.
       • Tweezers have two third class levers.
       • The top blade of a nail clipper is a third class-lever where the length of the load arm is only a bit longer than the effort arm.
       • Most swung items like swords, bats, hoes, brooms, maces, etc.
  • Inclined plane. MA = d _slope / d _lateral.
    • EGs:
      • A ramp.
    • Wedge. A portable double inclined plane.
    • Screw. A helical inclined plane that also also converts rotational force (torque) into linear force and vice versa.
  • Wheel and axel. MA = d _outer / d _inner.
    • If used for MA, then the wheel is a second-class lever.
    • If used for increase in speed, then the wheel is a third-class lever.
  • Pulley.
    • First-class pulley or fixed pulley. MA = 1. It does however change the direction of the pull. Also in real life it uses friction.
    • Second-class pulley or movable pulley. MA = 2.
    • Compound pulleys can achieve greater MA.
• Contrast the simple machines with gears. While both can provide MA, gears are more complex and usually focus on transmitting motion between shafts, changing direction of rotation, changing speed, changing between linear and rotational force.
• Conservation Laws [W]
  • Exact Conservation Laws
    • Conservation of energy
    • Conservation of linear momentum
    • Conservation of angular momentum
    • Conservation of electric charge
    • Conservation of color charge
    • Conservation of weak isospin
    • Conservation of probability
  • Approximate Conservation Laws
    • Conservation of mass (applies for low speeds)
    • Conservation of baryon number
    • Conservation of lepton number
    • Conservation of flavor (violated by the weak interaction)
    • Conservation of parity
    • Cosnervation of CP symmetry
• Laws of thermodynamics [W]
  • Zeroth law of thermodynamics: If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other.
  • First law of thermodynamics: In any process, the total energy of the universe remains the same. Aka The law of conservation.
  • Second law of thermodynamics: The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.
  • Third law of thermodynamics: As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.

Page Modified: (Hand noted: 2007-10-24 04:52:36Z) (Auto noted: 2010-12-24 22:58:50Z)