Classical Mechanics - a Brief Summary

[last updated: 2025-06-06]
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• Classical/Newtonian Mechanics:
  • Fundamental Quantities:
    These are the quantities that cannot be derived from other quantities:
    • mass (units: kg)
    • length (units: meter = m)
      This is a vector quantity, with both magnitude and direction.
    • time (units: seconds = sec)

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  • Derived Quantities:
    These are commonly-used quantities that can be derived from the fundamental quantities above:
    Note that any quantity derived from the fundamental quantity of length,
        which is a vector quantity,
    will also be a vector quantity.
    • velocity = length / time (units: m/sec)
      This is a vector quantity, since length is a vector with direction.
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    • acceleration = velocity / time (units: m/sec²)
      This is a vector quantity, with both magnitude and direction.
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    • Force = mass x acceleration (units: kg-m/sec² = newtons - N)
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    • Momentum (linear) (typ. symbol: rho = ρ) = mass x velocity (in a straight line) (units: kg-m/sec)
      This is a vector quantity, since velocity is a vector.
      The mass term can be considered as a measure of inertia or resistance to being moved/accelerated to the given velocity.

      
      Even if the mass that is being moved is distributed in some shape in space,
          it can be considered to be focused/condensed into a single point (its "center of mass"),
          and the equation holds, based on linear movement/velocity of that center of mass.

      a related quantity...

      Impulse = force x time
      This is the change in momentum that is imparted by an application of force on a body.

      = (mass x acceleration) x time
      = kg x m/sec² x sec
      = kg x m/sec
      (ie. units/dimension is identical to momentum)

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    • Momentum in a rotating body - Angular Momentum: (units: kg-m²/sec)
      • Angular momentum is typically denoted:   L
      • Angular momentum is more complicated than linear momentum because the movement/motion is not linear, but is a rotation around an axis.
        • The line of motion of any individual element of the rotating object is not linear,
          but follows a line that describes an arc around the axis of rotation, at some radius r from that axis.
          A key principle:
              Consider that a given element of mass that is farther from the axis of rotation will have a higher momentum
                  than an identical element of mass that is closer to the axis of rotation.
          As a consequence of this, r must be incorporated into the calculation.
          This is reflected in the fact that the MKS units for angular momentum are kg-m²/sec
              vs. kg-m/sec for linear momentum.
          The extra "m", or meter, is a result of including the length of the radius in the calculation.

        • A further complication is that different parts/elements of the object have different radii/distances from the axis of rotation,
          meaning you cannot treat the whole mass as being a point located at the object's center of mass,
              like you do with a linear momentum calculation,
          but instead must calculate the total angular momentum as a sum of the momenta of each infinitesimally-small-point element in the body.

      • The equation for calculating L is analogous to that for calculating linear momentum:
        a mass/inertia/resistance-to-acceleration term
        multiplied by a velocity term.
        Angular Momentum is a vector quantity, with the direction specified as the axis of rotation (right-hand rule)
          The equation for angular momentum:

        L (units: kg-m² / sec) = I x ω
        where: I = Moment of Inertia (units: kg-m²), and
            ω = angular velocity (units: sec^-1)

      • Moment of Inertia - the "mass" term in the angular momentum calculation: (units: kg-m²)
        • As stated above, because the momentum contributed by an individual element of the rotating object
          is affected by the distance of that element from the axis of rotation, that r distance from the axis must be accommodated in the calculation.
        • This is done with the quantity of Moment of Inertia. (typ. symbol: I )
          The I is a property of the object as a whole - including the chosen axis of rotation.

          It represents a measure of resistance of the object to rotational acceleration around a given axis.
          I is the sum (integral) of mass x r² for all the elements in the object.

        • Note that different shaped objects will have different I's, even if their total masses are the same,
          and the same object will have different I's, when calculated using different axes of rotation.
          Moment of Inertia for some common shapes:
          • Simple/trivial case of a point-mass located some r from axis of rotation:
            I = M x r²
            where: M = object's mass
            Note: this is the base equation used to calculate MOI for an object of any shape.
            The total MOI for a complex-shaped object
            is just the sum/integral of MOI (M x r2) for each mass element in the object.
          • sphere with axis through its center:
            I = 2/5 x M x r²
            where: M = object's mass, and
                r = radius of sphere
          • flat, circular plate (a disc), with axis through its center (perpendicular to the plate):
            I = 1/2 x M x r²
            where: M = object's mass, and
                r = radius of disc

      • The "velocity" term in the angular momentum calculation:
        • Rotational or angular velocity is expressed in radians-of-rotation/sec.
          where a single, full rotation is 2-pi radians
        • Angular velocity (typ symbol: ω)
        • Radians:
          • Radians are used to measure angles, and strictly are ratios:
            radians = arc-length / radius
          • For example: Calculate the number of radians in a 360 deg arc (ie. a full circle):
            The arc-length of a full circle is its circumference, which is:
            π x diam = π x 2 x radius = 2 x π x r

            So: radians = arc-length / radius = (2 x π x r) / r = 2 x π

          • Note that radians are dimensionless.
            That is, they are a ratio of arc-length / radius-length, or meters/meters, or unity dimensionless.
        • Since angular velocity is radians/sec, and radians are dimensionless,
          units of angular velocity are sec^-1

      • recap: Angular Momentum:
        L = I x ω
            = (kg-m²) x (sec^-1)
            = kg-m² / sec

      • Angular Momentum is a "conserved" quantity.
        Once a given system is placed into motion, it's angular momentum will stay the same.
        For example, an ice skater in a spin will have some angular momentum, determined by her moment of inertia and her rotational speed..
        If she moves to bring her arms closer to her body, then her moment of inertia will be less,
        and her rotational speed (her angular velocity) will (must) increase in order to maintain a constant angular momentum.

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    • Energy (= Work) (units: kg-m²/sec² = J = N-m)
      • Energy is force applied over some distance
        Energy = force x distance
            = (kg-m/sec²) x (meters)
            = kg-m²/sec²
            = Joules
      • Kinetic Energy:
        KE ("moving energy") = (mass x velocity²) / 2
            = kg x (m/sec)²
            = kg-m²/sec²
      • Potential Energy: (U)
        • Gravitational PE is the "stored" energy in an object that is elevated above ground
          that would be released into KE if the object were "dropped" and allowed to fall to the ground.
          It is the result of the force of gravity pulling on the object.
              U = M x g x h
                  where: M = mass of the object
                  g = acceleration of gravity from the Earth
          • The acceleration of gravity from the earth is treated as a constant = 9.8 m/sec²
            though this is only accurate to about 1% near the Earth's surface.

                  h = height above Earth

                  U units = kg x m/sec² x m
                  = kg-m²/sec²

        • There are 6 other types of potential energy: elastic, electric, chemical, magnetic, heat, nuclear.

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    • Power:
      • Power is Work (or energy) per unit time:
        Power = Joules / sec
            = (kg-m²/sec²) / sec
            = kg-m² / sec³
            = watts