# 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 V
_{t}and V_{n}be*ds'*s velocities, - Of which V
_{t}along and V_{n}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 V_{t}.- V
_{n}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 V
_{n}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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