Matthew van Eerde's web log

  • Matthew van Eerde's web log

    Deriving the centripetal acceleration formula

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    A body that moves in a circular motion (of radius r) at constant speed (v) is always being accelerated.  The acceleration is at right angles to the direction of motion (towards the center of the circle) and of magnitude v2 / r.

    The direction of acceleration is deduced by symmetry arguments.  If the acceleration pointed out of the plane of the circle, then the body would leave the plane of the circle; it doesn't, so it isn't.  If the acceleration pointed in any direction other than perpendicular (left or right) then the body would speed up or slow down.  It doesn't.

    Now for the magnitude.  Consider the distance traveled by the body over a small time increment Δt:

    http://blogs.msdn.com/photos/matthew_van_eerde/images/9952566/original.aspx

    We can calculate the arc length s as both the distance traveled (distance = rate * time = v Δt) and using the definition of a radian (arc = radius * angle in radians = r Δθ:)

    http://blogs.msdn.com/photos/matthew_van_eerde/images/9952564/original.aspx

    The angular velocity of the object is thus v / r (in radians per unit of time.)

    The right half of the diagram is formed by putting the tails of the two v vectors together.  Note that Δθ is the same in both diagrams.

    http://blogs.msdn.com/photos/matthew_van_eerde/images/9952565/original.aspx

    Note the passing from sin to cos is via l'Hôpital's rule.

    QED.

  • Matthew van Eerde's web log

    Newton: combining celestial and terrestrial mechanics

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    For as long as there have been humans, we have looked at the skies - and around us on Earth as well.  Things move in the skies; things move on Earth.  The way things move in the skies is called "celestial mechanics"; the way things move on Earth is called "terrestrial mechanics."

    The ancient Greeks thought there were five elements, corresponding to the five regular (Platonic) solids: the four terrestrial elements (air, fire, earth, and water) and the fifth celestial element which the Romans called quintessence.

    All kinds of rules were discovered/proposed for how things move on Earth.  Galileo demonstrated that falling objects accelerate at the same rate regardless of their mass.  Similar rules were discovered/proposed for how things move in the heavens.  Kepler demonstrated that the planets move in ellipses around the sun, that they swept out equal areas in equal times, etc.  But everybody knew that things in the heavens were different than they were on Earth.

    Then Newton, building on Hooke, came up with the crazy idea that perhaps things fall on Earth for the same reason that heavenly objects go in ellipses... that all objects, celestial or terrestrial, act on each other from a distance in a uniform fashion.  (He himself wrote that this was such a crazy idea that no-one should take it seriously.)

    That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it. -- Isaac Newton

    But he had done some calculations and it seemed to work out... specifically, he worked out how much acceleration you would expect the Earth to exert on an object as far away as the Moon, and how much acceleration the Moon would need to stay in its orbit (and not wander off or spiral down into the Earth.)

    I... compared the force requisite to keep the moon in her orb with the force of gravity at the surface of the earth and found them to answer pretty nearly. -- Isaac Newton

    Note Newton's use of the word "orb" to describe the motion of the moon, a direct reference to celestial mechanics.

    Let's see how compelling those calculations were, shall we?

    His first idea was that there was this thing called "force" which was equal to the mass of an object multiplied by the acceleration it was undergoing.  F = m a.  Fine.

    His second idea (which he stole, at least partially, from Hooke) was that any two objects have a force that attracts them, which is proportional to both of their masses, and inversely proportional to the square of the distance between them (that's the part he stole from Hooke.)  F = G m1 m2 / r2.  Fine.

    His third idea was that an object that goes around a circle of radius r at a constant speed v is undergoing a constant centripetal acceleration of a = v2 / rThis follows pretty quickly from the definitions with a little geometry, trigonometry, and a dash of calculus.  Fine.

    He also knew some facts.  I'm going to state them in metric because, hey, this is the twenty-first century.  He knew:

    • The distance from the center of the Earth to the surface (where the apple trees are) is 3,960 miles: 6370 km.
    • Falling objects on the surface of the Earth accelerate at 32.2 ft/s2: 9.81 m/s2.
    • The distance from the center of the Earth to the center of the Moon is 239,000 miles: 384,000 km.
    • The Moon takes 27.3 days to make a single sidereal orbit around the Earth: 236,000 s.

    http://blogs.msdn.com/photos/matthew_van_eerde/images/9952568/original.aspx

    Coarse estimate time...

    If the Moon is roughly 50 times further away from the center of the Earth than an apple is, and "gravitational" acceleration is inversely proportional to the square of the distance between two objects, then the Moon should be accelerated roughly 1/2500th as much as the apple is: 0.004 m/s2 instead of 10 m/s2,

    More precisely...

    Consider the apple case (mE is the mass of the Earth, mA the mass of the apple:)

    FA = mA aA = G mE mA / rA2
    a
    A rA2 = G mE
    9.81 m/s2 (6.37e6 m)2 = G mE
    G
    mE = 3.98e14 m3/s2

    Now consider the moon case:

    FM = mM aM = G mE mM / rM2
    aM = G mE / rM2
    aM = 3.98e14 m3/s2 / (3.84e5 m)2
    aM = 0.00269 m/s2

    So Newton's proposed formula predicts that the Earth's gravity causes the Moon to accelerate at 0.00269 m/s2.  This is, indeed, roughly 0.004 m/s2 so it jives with our coarse estimate.

    Let's see how that compares to the necessary centripetal acceleration to achieve a closed orbit: a = v2 / r.

    The distance the moon travels in its sidereal orbit is 2π * 3.84e5 m = 2.42e9 m, and it does so in 27.3 days; so its velocity is 2π * 3.84e5 m / (27.3 days * 24 (hours / day) * 60 (minutes / hour) * 60 (s / minute)) = 1020 m/s.  This is roughly Mach 3 in our atmosphere.

    The centripetal acceleration is thus measured to be (1020 m/s)2 / 3.84e8 m = 0.00273 m/s2. Newton's prediction is about 1.5% off, which isn't bad, considering.

    Note that although Newton knew the value of the product G mE, he didn't know either G or mE seperately.  It wasn't until Cavendish took two known masses and measured the gravitational pull between them directly that G was evaluated; we could then calculate the mass of the earth as G mE / G.

    Once G was known we could also evaluate the mass of the Sun based on Earth's orbit.  However, we still could not determine the mass of the Moon... in fact, back in the 1950s, American scientists were unable to determine the mass of Sputnik (which would have been very helpful) even though we could see its orbit.

    Which raises the question... how did we know the mass of the Moon?

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January, 2010