A Problem in Dynamics
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- AN inextensible heavy chain
- Lies on a smooth horizontal plane,
- An impulsive force is applied at A,
- Required the initial motion of K.
- Let ds be the infinitesimal link,
- Of which for the present we've only to think;
- Let T be the tension and T + dT
- The same for the end that is nearest to B.
- Let a be put, by a common convention
- For the angle at M 'twixt OX and the tension;
- Let Vt and Vn be ds's velocities,
- Of which Vt along and Vn across it is;
- Then the tangent will equal,
- Of the angle of starting worked out in the sequel.
- In working the problem the first thing of course is
- To equate the impressed and effectual forces.
- K is tugged by two tensions, whose difference dT
- [1] Must equal the element's mass into Vt.
- Vn must be due to the force perpendicular
- To ds's direction, which shows the particular
- Advantage of using da to serve at your
- Pleasure to estimate ds's curvature
- For Vn into mass of a unit of chain
- [2] Must equal the curvature into the strain.
- Thus managing cause and effect to discriminate,
- The student must fruitlessly try to eliminate,
- And painfully learn, that in order to do it, he
- Must find the Equation of Continuity.
- The reason is this, that the tough little element,
- Which the force of impulsion to beat to a jelly meant,
- Was endowed with a property incomprehensible,
- And was "given", the the language of Shop, "inextensible."
- It therefore with such pertinacity odd defied
- The force which the length of the chain would have modified,
- That its stubborn example may possibly yet recall
- These overgrown rhymes to their prosody metrical.
- The condition is got by resolving again,
- According to axes assumed in the plane.
- If then you reduce to the tangent and normal,
- [3] You will find the equation more neat tho' less formal.
- [4] The condition thus found after these preparations,
- When duly combined with the former equations,
- Will give you another, in which differential
- [5] (When the chain forms a circle), become in essentials
- No harder than those that we easily solve
- [6] In the time a T totum would take to revolve.
- Now joyfully leaving ds to itself, a—
- Ttend to the values of T and of a.
- The chain undergoes a distorting convulsion,
- Produced first at A by the force of impulsion.
- In magnitude R, in direction tangential,
- [7] Equating this R to the form exponential,
- Obtained for the tension when a is zero,
- It will measure the tug, such a tug as the "hero
- Plume-waving" experienced, tied to the chariot.
- But when dragged by the heels his grim head could not carry aught,
- [8] So give a its due at the end of the chain,
- And the tension ought there to be zero again.
- From these two conditions we get three equations,
- Which serve to determine the proper relations
- Between the impulse and each coefficient
- In the form for the tension, and this is sufficent
- To work out the problem, and then, if you choose,
- You may turn it and twist it the Dons to amuse.
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This work was published before January 1, 1929, and is in the public domain worldwide because the author died at least 100 years ago.
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